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CO 00 

<OU_1 64333 



M.A., D.Sc., LL.D., F.R.S. 

Plumian Professor of Astronomy and Experimental 
Philosophy in the University of Cambridge 






^Preface page vi 

Chapter I Science and Experience I 

II Dramatis Personae 27 

Til The End of the World 50 

IV The Declina'ofT^terminism 72 

V Indeterminacy and Quantum Theory 92 

VI Probability no 

VII The Constitution of the Stars 135 

VIII Subatomic Energy 160 

IX Cosmic Clouds and Nebulae 184 

X The Expanding Universe 206 

XI The Constants of Nature 229 

XII The Theory of Groups 255 

XIII Criticisms and Controversies 278 

XIV Epilogue 309 
Index 327 


Plate i Electrons and Positrons Facing page 28 
By permission of Prof. P. M. S. Blackett 

2 Gaseous Nebula (Cygnus) 184 

3 Dark Nebulosity The Horse's Head 202 

4 Spiral Nebula (Canes Venatici) 206 


THIS volume contains the Messenger Lectures which I 
delivered at Cornell University in April and May 1934. 
Chapters n and vin have been added ; the remaining chapters 
correspond to the twelve lectures of the course. It was one 
of the conditions of the lectureship that the lectures should 
be published. 

Except for a small book on the Expanding Universe, my 
last spell of writing was about six years ago, when Stars and 
Atoms (1927), The Nature of the Physical World (1928) and 
Science and the Unseen World (1929) practically exhausted all 
that it was then in my mind to say. A scientific writer is 
placed in a difficulty by his earlier books ; either his new book 
will appear as a rather disjointed addendum to them, or he 
must perfunctorily go over again a great deal of matter 
which he has no wish to rewrite. Being unwilling to adopt 
the second alternative, I determined to make what I could 
of whatever had come to my mind in the last six years. 
Accordingly I spoke at Cornell on a variety of topics, using 
as a nucleus the material contained in a number of addresses 
and lectures which I had had occasion to deliver since 1929, 
and adding other subjects to which I had been giving attention. 
The general plan was that each lecture should have a separate 
theme, except that Indeterminism was spread over two 
lectures. The choice of subjects has allowed a certain amount 
of continuity of treatment; but there has been no attempt 
to provide a systematic introduction to modern scientific 
thought. Perhaps the biggest gap is the absence of any 
account of the elementary ideas of the theory of relativity; 


I could not bring myself to go over again the ground covered 
in Chapters I, n, m, vi, vn of The Nature of the Physical World 
altering the treatment and illustrations merely for the sake 
of alteration. 

In the opening lecture I try to explain the philosophical 
outlook of modern science, as I understand it, and show how 
the scientific picture of the world described in physics is 
related to the "familiar story" in our minds. Chapter n is 
an interpolation containing a summary of our knowledge 
of atomic physics, etc. , which some readers may find necessary 
for an understanding of subsequent chapters and others may 
find useful as a reminder. Then follow four lectures which 
have something in common; they are concerned with the 
consequences of the statistical type of law, first introduced 
into physics in the subject of thermodynamics, which has in 
recent years completely driven out the older causal type of 
law from the foundations of physics. The last of these four 
lectures, on Probability, has besides its application to statistical 
law a more elementary interest. 

Then follows a complete change of subject, and the next 
four lectures are devoted to astrophysics. Starting with the 
sun and familiar stars, we advance to greater distances till we 
reach the system of milliards of galaxies which constitutes 
the universe. This last subject has been treated more fully in 
my recent book The Expanding Universe; I here give a much 
shorter account. In this lecture (Chapter x) we meet the 
elusive "cosmical constant" which takes us back to the 
fundamental conceptions of physics again for the next two 
chapters. Chapter xi is, I realise, much too severe for this 
kind of book; I can only plead that the subject which has 
occupied me for the last five years, almost to the exclusion 
of any other research, was bound to spill over into any course 


of lectures I might give. The next lecture, on Theory of 
Groups, was something of an experiment; but it, more 
nearly than any other part of the book, touches the key-note 
of scientific philosophy. 

The chapter "Criticisms and Controversies" may by its 
title lead the reader to expect a comprehensive series of 
answers to the multitudinous points raised by critics and 
reviewers, and by many who have contributed valuable dis- 
cussion of the views which I have advocated. I think that a 
little reflection will show that this was impracticable with 
any reasonable allotment of space. If a criticism can be 
answered briefly and decisively it seems scarcely worth while 
to inform the world in general that so-and-so has raised it. 
If it is more arguable, a lengthy explanation and discussion 
of it is usually necessary. For the most part I am content to 
think that if my contentions are of value they will find their 
proper level without continual parental intervention to save 
them from determined opponents and sometimes from 
over-enthusiastic friends. But I would express here rny 
gratitude for many articles by philosophers and others 
courteously discussing my writings. Sometimes I have ap- 
preciated the justice of the criticism, and it has had its due 
influence in maturing my views. Often I would have liked 
to write a reply in the hope of advancing an understanding 
on both sides; but such a reply requires at least as much time 
and care as an independent article, and with rare exceptions I 
have had to let the opportunity go by. In the concluding 
lecture I return again to the philosophical outlook of 
Chapter I, but tijis time I refer to that part of " the problem 
of experience" which the methods of physics do not profess 
to treat. Parts of this lecture are taken from an address which 
I gave in a broadcast symposium on Science and Religion. 


As usual, notwithstanding my efforts to simplify things, 
I have to impose a rather heavy strain on the attention of the 
reader. Since the chapters are to a considerable extent in- 
dependent, the difficulty tends to increase towards the ends 
of the chapters. There is hope of a respite when the next 
chapter begins. 

These lectures carry for me happy memories of the weeks 
which I spent in Cornell University. To the friends who 
welcomed me, and to the large audiences who encouraged 
me, I dedicate them gratefully. 

A. S. E. 


September 1934 


Docs the harmony which human intelligence thinks it discovers in Nature 
exist apart from such intelligence ? Assuredly no. A reality completely 
independent of the spirit that conceives it, sees it or feels it, is an im- 
possibility. A world so external as that, even if it existed, would be for 
ever inaccessible to us. What we call "objective reality" is, strictly 
speaking, that which is common to several thinking beings and might 
be common to all; this common part, we shall see, can only be the 
harmony expressed by mathematical laws. 

POINCAR, The Value of Science. 


Asa conscious being I am involved in a story. The perceiving 
part of my mind tells me a story of a world around me. The 
story tells of familiar objects. It tells of colours, sounds, 
scents belonging to these objects; of boundless space in which 
they have their existence, and of an ever-rolling stream of rime 
bringing change and incident. It tells of other life than mine 
K.tcy about its own purposes. 

Ls a scientist I have become mistrustful of this story. In 
(many instances it has become clear that things are not what 
they seem to be. According to the story teller I have now in 
front of me a substantial desk ; but I have learned from physics 
that the, desk is not at all the continuous substance that it is 
supposed to be in the story. It is a host of tiny electric charges 
darting hither and thither with inconceivable velocity. 
Instead of being solid substance my desk is more like a swarm 
of gnats. 

So I have come to realise that I must not put overmuch 
confidence in the story teller who lives in my mind. On the 
other hand, it would not do to ignore him altogether, since 
his story generally has some foundation of truth more 



especially in those anecdotes that concern me intimately. 
For I am given a part in the story, and if I do not take my 
cue with the other actors it is the worse for me. For example, 
there suddenly enters into the story a motor car coming 
rapidly towards the actor identified with myself. As a 
scientist I cavil at many of the particulars given by the story 
teller the substantiality, the colour, the rapidly increasing 
size of the object approaching- but I accept his suggestion 
that it is wisest to jump out of the way. 

There are ponderous treatises on my shelves which tell 
another story of the world around me. We call this the 
scientific jftory. One of our first tasks must be to try to 
understand the relation between die familiar story and the 
scientific story of what is happening around us. 

At one time there was no very profound difference between 
the two versidns. The scientist accepted the familiar story in 
its main outline; only he corrected a few facts here and there, 
and elaborated a fe^/ details. But latterly the familiar story 
and the scientific story have diverged more and more widely 
until it has become hard to recognise that they have any- 
thing in common. Not content with upsetting fundamentally 
our ideas of material substance, physics has played strange 
pranks with our conceptions of space and time. Even 
causality has undergone transformation. Physical science now 
deliberately aims at presenting a new version of the story of 
our experience from the very beginning, rejecting the familiar 
story as too erratic a foundation. 

But although we try to make a clean start, rejecting 
instinctive or traditional interpretations of experience and 
accepting only the kind of knowledge which can be inferred 
by strictly scientific methods, we cannot cut ourselves loose 
altogether from the familiar story teller. We lay down the 
principle that he is always to be mistrusted; but we cannot 
do without him in science. What I mean is this: we rig up 
some delicate physical experiment with galvanometers, 


micrometers, etc., specially designed to eliminate die fallibility 
of human perceptions; but in the end we must trust to our 
perceptions to tell us the result of the experiment. Even if 
the apparatus is self-recording we employ our senses to read 
the records. So, having set the experiment going, we turn 
to the familiar story teller and say "Now put that into your 
story". He has perhaps just been telling us that the moom is 
about the size of a dinner plate, or something equally crude 
and unscientific; but at our interruption he breaks off to 
inform us that there is a spot of light coinciding with division 
No. 53 on the scale of our galvanometer. And this time we 
believe him more or less. At any rate we use this informa- 
tion as die basis of our scientific conclusions. If we are to 
begin actually at the beginning we must inquire why we 
trust the story teller's information about galvanometers in 
spite of his general untrustworthiness. For presumably his 
tertile invention is quite capable of "embroidering" even a 

I do not want to spend time over points which no scienti- 
fically-minded person disputes; so I will assume that you 
agree that die only channel of communication between the 
story teller who lives in your mind and the external world 
which his story professes to describe is the nervous system 
in your body. In so far as your familiar conception or picture 
of what is going on around you is founded on your sense of 
sight, it depends on impulses transmitted along the optic 
nerves which connect the retina with the brain. Similarly 
for your other sense organs. You do not, of course, perceive 
the impulses themselves; the story teller has worked them 
up into a vivid story. The inside of your head must be rather 
like a newspaper office. It is connected with the outside 
world by nerves which play the part of telegraph wires. 
Messages from the outside world arrive in code along these 
wires ; the whole substratum of fact is contained in these <;ode 
messages. Within die office they are made up into a pre- 



sentable story, partly by legitimate use of accumulated 
experience but also with an admixture of journalistic 
imagination; and it is this free translation of the original 
messages that our consciousness becomes aware of. 

If we had a complete record of the impulses transmitted 
along the nerves we should have all the material which the 
story teller can have had as a foundation for his story in so 
far as his story relates to the external world. And it is to this 
material that we must appeal if we wish to discover the truth 
behind the story. To appreciate the task of physical science 
let us then suppose that we are in possession of these data 
the dots and dashes, or whatever the signals are, that arrive 
at the brain cells at the terminations of the nerves. All that 
physical science can assert about the external world must be 
inferable from these. If there is any part of our conception 
of the physical universe which cannot have come to us in the 
form of nerve signals we must cut it out. As in a beleaguered 
city there spread circumstantial rumours of happenings in 
the world outside which cannot have been received from 
without, so in our minds there arise all sorts of conceptions 
of entities and phenomena in the external world which 
cannot have been transmitted to us from outside. They do 
not conform to the type of message which the narrow threads 
of communication will bear. We are continually making the 
mistake of the man who, on receiving a telegram, thinks 
that the handwriting is that of the sender. The messages as 
we become aware of them in consciousness are dressed up 
with conceptions of colour, spatiousness, substance. This 
dress is no part of the message as it was handed in by the 
external universe. It is assumed after the message arrives; 
for the transmitting mechanism is by its very nature incapable 
of conveying such forms of conception. 

This limitation of the transmitting mechanism is strikingly 
illustrated when we talk with a colour-blind person. We 
know from his amazing mistakes that there is a big difference 


between his perception of his surroundings and ours. But 
he is quite unable to convey to us how his perception differs. 
When he confuses red with green, does he see both colours 
as red or both as green or as some hue unknown to us? He 
has no means of telling us. The intrinsic nature of his per- 
ception is trapped in his mind. It cannot flow out along his 
nerves; nor could it travel up our nervous system if it reached 
it. Similarly the sensory qualities of colour, sound and scent 
cannot have been transmitted to us from the object in the 
external world to which we attribute the colour, sound and 
scent; for even if we suppose the object itself to be endowed 
with such qualities it would be as impotent as the colour- 
blind person to convey to us their character. The part played 
by the external object is to condition directly or indirectly 
the signals which pass along the nerves. The story which 
arises in our consciousness is a consequence of these signals, 
but it contains much that does not belong to the external 

The inference of any kind of knowledge of the physical 
objects which he at the far end of these lines of communica- 
tion must evidently be very indirect. In this respect it differs 
from the knowledge constituted by the mind's immediate 
awareness of its own sensations, thoughts, emotions. I have 
elsewhere expressed this in the words: "Let us not forget 
that mind is the first and most direct thing in our experience; 
all else is remote inference''.* That is a statement which, 
I believe, physicists accept almost as a truism, and philo- 
sophers generally condemn as a hoary fallacy. It is difficult 
to understand why there should be such a difference between 
us. I had thought that, like many other differences, it might 
arise because we do not talk the same language; but some 
recent writings seem to show that the cleavage may be 
deeper, and that there is a tendency in modern philosophy 
to adopt a view which is scientifically untenable, f 
* Science and the Unseen World, p. 24. f See pp. 280-288. 


Scientific thinkers generally agree that the channel of 
communication between the external world and man's con- 
sciousness is severely limited in this way; but, whilst giving 
intellectual assent, they do not always adjust their scientific 
outlook to correspond. They are strangely reluctant to doubt 
the assertions of the familiar story teller even when it is 
evident that he is talking through his hat. The feeling that 
many of the conclusions of relativity theory and quantum 
theory are contrary to common sense is largely due to this 
tenacity. We cling to certain features in the familiar picture 
of the external world, almost as though we were persuaded 
that some part of our percipient selves had been projected 
outside the body, and had entered into external things and 
become aware of their ultimate nature in the same direct way 
that the mind is aware of its thoughts and sensations. We 
uphold the familiar conceptions of space in the external world 
as assuredly as if the spirit of man could enter into space and 
feel what it is like to be large or small. But when an external 
object raps on the door at the extremity of a nerve, you can- 
not put your head outside to see what is rapping. You cannot 
know more of its nature than that it must be such as to 
account for the delivery of the raps in their sequence. 
A scientific theory which accounts for the raps is none the 
worse because it runs counter to the story teller's habitual 
but unwarranted picture of what lies beyond the ever-sealed 


Broadly speaking the task of physical science is to infer 
knowledge of external objects from a set of signals passing 
along our nerves. But that rather underrates the difficulty 
of the problem. The material from which we have to make 
our inferences is not the signals themselves, but a fanciful 
story which has been in some way based on them. It is as 
though we were asked to decode a cipher message and were 


given, not the cipher itself, but a mistranslation of it made 
by a clumsy amateur. 

It is true that the physiologist nowadays is able to tap the 
messages as they pass along a nerve. He can record the 
changes of electrical potential that occur when a nerve is 
stimulated, and the record shows a series of oscillations which 
are presumably the physical foundation for the perception 
that arises in the mind. But we cannot begin the study of 
the external world with these records. In order to utilise 
them a rather advanced scientific knowledge of the nature 
of the human body and the functions of the various nerves 
is presupposed. All that the physiologist has done is to tap 
the messages on the way to one brain and divert them into 
another brain his own. That is not fundamentally different 
from the method of the physicist who intercepts the messages 
emanating from physical objects before they reach any 
nerve, and, for example, causes them to record themselves 
on a photographic plate. By one route or another the 
messages must ultimately be conducted to a seat of con- 
sciousness if they are to be translated into knowledge. 

It is the inexorable law of our acquaintance with the 
external world that that which is presented for knowing 
becomes transformed in the process of knowing. 

Thus in saying that the initial data of physics are nerve 
signals, we must not be confused by the fact that nerve 
signals are pictured by us as known processes in the external 
world. This identification of our initial data is not itself an 
initial datum; it is one of our indirect inferences. It all 
emphasises the difficulty of tracing our knowledge of the 
physical world to its beginning. We detect it stealing into 
our minds through our nerves; but our knowledge of the 
physical world had to be considerably advanced before we 
discovered that we possessed a nervous system. 

More by the exigencies of its own development than by 
the consideration that we h^ve been discussing, modern 


physics has been forced to recognise the gulf between the 
external world which appears in the familiar story of per- 
ception and the external world which presents its messages 
at the door of the mind. It is for this reason that the scientific 
story is no longer a tinkering of the familiar story but follows 
its own plan. I think the modern view can best be expressed 
by saying that we treat the familiar story as a cryptogram. 

Our sensory experience forms a cryptogram, and the 
scientist is a Baconian enthusiast engaged in deciphering the 
cryptogram. The story teller in our consciousness relates a 
drama let us say, the Tragedy of Hamlet. So far as the drama 
is concerned the scientist is a bored spectator; he knows the 
unreliability of these play-writers. Nevertheless he follows 
the play attentively, keenly alert for the scraps of cipher that 
it contains; for this cipher, if he can unravel it, will reveal 
a real historical truth. Perhaps the parallel is closer than 
I originally intended. Perhaps the Tragedy of Hamlet is not 
solely a device for concealing a cryptogram. I would admit 
nay, rather I would insist that consciousness with its 
strange imaginings has some business in hand beyond the 
comprehension of the cipher expert. In the truest sense the 
cipher is secondary to the play, not the play to the cipher. 
But it is not our business here to contemplate those attributes 
of the human spirit which transcend the material world. We 
are discussing the external world of physics whose influences 
only reach us by signals along the nerve fibres; and so we 
have to deal with the story after the manner of a cryptogram. 

The solution of a cryptogram is found by studying the 
recurrency of the various signs and indications. I do not think 
we should ever have made progress with the problem of 
inference from our sensory experience, and theoretical physics 
would never have originated, if it were not that certain 
regularities and recurrencies are noticeable in sensory ex- 
perience. We call these regularities of experience laws of 
Nature. When such a law has been established it becomes also 


a rule of inference, so that it helps us in further decipherment 
just as in solving an ordinary cryptogram. 

I do not know how a logician would classify the process 
of solving a cryptogram. The decoded message is inferred 
from the cryptogram, but the method of inference can 
scarcely be described as logical deduction. In saying that the 
scientific description of the external world is inferred from 
our sensory experience, and that the entities of the physical 
world are inferences, I use the word inference in this broad 

Our task then is to discover a scheme revealed by the 
regularities and recurrencies in our sensory experience. Since 
these regularities occur in the sensory experience of all men 
the scheme is presented as an external world linking together 
the experiences of different individual consciousnesses. In 
thus defining the object of our search we determine to a 
certain extent the nature of that which we shall find. The 
universe of physics must by its very definition have the two 
characteristics of regularity (or partial regularity) and ex- 
ternality. We do not contest the right of anyone who is 
interested in other aspects of sense data, or of the conscious- 
ness in which they reside, to pursue his investigations in his 
own way; but so far as physical science is concerned we drop 
everything that is inessential to the elucidation of regularities 
and recurrencies. 

I must also emphasise the significance of the term 
" external". The familiar world of my perception seems to 
be external; but, in the courts of science, what the familiar 
story teller says is not evidence. The world of my dreams 
also seems to be external, but it has no existence outside my 
mind. The argument that the world containing the entities 
of physics is external is quite independent. When I examine 
the content of my consciousness with a view to formulating 
the recurrencies of my sensory experience, there are two 
possible ways of treating the data two ways in which I 


might attempt to solve the cryptogram. Among the data 
are certain auditory sensations "spoken words'* and certain 
visual sensations "printed words" which admit of alternative 
treatment. I might study their recurrencies and regularities 
without discriminating them from other auditory and visual 
sensations. Then all the recurrencies are of data within my 
own consciousness and the study of them never takes me 
outside the region of my own mind; the solution of the 
cryptogram, if any, reached by this treatment will be an 
internal egocentric world. But such a treatment of the 
problem of experience is not often promulgated if only 
because a lecturer cannot deny himself the hope that his 
"spoken words" wiH be treated by his audience as on a 
different epistemological footing from the beating of a tin 
can. Therefore in science and in most philosophies spoken 
and printed words are treated, not only as immediate sensory 
data of our own consciousness, but as communicating to us 
data existing in other consciousnesses. 

Thus our first intimation of externality has no direct 
connection with physical science. It comes from the recogni- 
tion that the problem of experience is concerned with data 
distributed among many different individual consciousnesses. 
The synthesis of experience then necessarily leads to the 
contemplation of a neutral domain not coextensive with any 
individual mind. Thus although we start from individual 
mental data, as soon as we commit ourselves to the recogni- 
tion of other minds than our own, we are led to the con- 
ception of an external domain (physical space and time) to 
contain the inferential objects of our combined knowledge. 
Among these inferential objects are the nerve fibres and 
brain cells where (as the decipherment of the cryptogram 
progresses) the sources of communication between the 
objects of this external world and an individual consciousness 
are found to be located. 

We asked why the story teller should be believed when he 


talks about galvanometers, although he is untrustworthy 
when he talks of familiar objects. I think the answer is that 
the truth of the story is not the point in question; the physicist 
is concerned only with the scraps of cipher contained in it. 
The galvanometer is a device for leading the story into 
situations in which the underlying cipher becomes less 
baffling to interpret; it is not a bridle on the story teller's 


A feature of progress in unravelling the cryptogram has been 
that much of our sense data proves to be redundant 
redundant, that is to say, in the study of recurrencies. We 
can, for example, drop the sense of hearing, since it only 
indicates regularities which can alternatively be detected by 
our other senses. With the reduction of the number of types 
of sense data to a minimum there has been a parallel unifica- 
tion of the external world. One scheme of regularity suffices, 
instead of a distinct scheme for each of our senses, with perhaps 
additional schemes corresponding to electric and magnetic 
senses which we presumably might have possessed if Nature 
had so chosen. This dropping of a variety of types of sense 
data is responsible for some of the most striking differences 
between the familiar and the scientific conception of the 
external world. 

Writing this chapter on an autumn day, I feel myself in 
a familiar world whose most prominent characteristic is 
colour. There is no colour in the physical world. I think 
that that is the right way to put it. It is true that each colour 
is represented in the physical world by a number supposed 
to indicate the length of a wave of some kind. Similarly 
I am represented at the telephone exchange by a number 
indicating a hole in a switch-board; but it would not be 
correct to say that I inhabit the telephone exchange. To put. 


it another way, there is nothing in the accepted description 
of the physical world which owes its acceptance to the fact 
that we have a sense of colour. Everything that we assert 
can be verified by a colour-blind person; and indeed most 
of our accurate knowledge has been ascertained through the 
medium of a colour-blind photographic plate. 

When we have eliminated all superfluous senses, what 
have we left ? We can do without taste, smell, hearing, and 
even touch. We must keep our eyes or rather one eye, for 
there is no need to use our faculty of stereoscopic vision. The 
eye need not have the power of measuring or graduating 
light and shade; I think it is sufficient if it can just discriminate 
two shades so as to detect whether an opaque object is in a 
certain position or not. 

With this reduced equipment we can still recognise 
geometrical form and size. We can recognise that one object 
appears round and another square, or that one is apparently 
larger than another. Some years ago the position had been 
reached that spatial form and magnitude were the only 
features in our familiar picture which existed also in the 
external world of science. This led to a geometrisation of 
physics. You can see why at this stage physics became so 
largely geometrical in its methods and vocabulary. The 
preserved data which contained the recurrencies, and there- 
fore the key to the cryptogram, were wholly geometrical; 
all other data had been dropped as redundant when it was 
found that they revealed only the same recurrencies as the 
geometrical data. 

By limiting the sensory equipment of our observers we 
do a great deal to stop their quarrelling. For example, by 
removing their ears we put an end to the disputes of the 
musical critics. I do not say that they are disputing about 
nothing; but, whatever it is, it is not relevant to the scheme 
of regularities of which the physicist is in search. 

But it was found that the observers were still quarrelling 


even when they had only form and size to quarrel over.* 
So in 1915 Einstein made another raid on their sensory 
equipment. He removed all the retina of the eye except one 
small patch. The observer could no longer recognise form 
or extension in the external world, but he could tell whether 
two tilings were in apparent coincidence or not. 

If you read about Einstein's theory of relativity you will 
find many references to a peculiar person called "the ob- 
server" the man who has a habit of falling down lifts, or 
getting transported by aeroplanes travelling at 161,000 miles 
a second. Now you have a picture of him. He has one eye 
(his only sense organ) which is colour-blind. He can dis- 
tinguish only two shades of light and darkness so that the 
world to him is like a picture in black and white. The sensitive 
part of his retina is so limited that he can sec in only one direction 
at a time. We allow him any number of assistants equipped 
like himself so that they can keep watch on the different 
parts of an experiment and pool their knowledge afterwards. 
Since we have so mutilated him he cannot make the experi- 
ments himself. We perform the experiments, and let him 
keep watch. The point is that all our knowledge of the 
external world as it is conceived to-day in physics can be 
demonstrated to him. If we cannot convince him we have 
no right to assert it. 

I will not stop to justify in detail this drastic method of 
inculcating respect for truth. I will only point out that it is 
not too intrinsically absurd, because we have left the observer 
power to recognise that a pointer coincides with a graduation 
on a scale. Practically every physical measurement which 
has any pretension to accuracy resolves itself into a pointer- 
reading of this kind. Instead of relying on our sense of 
warmth we read the graduation of a thermometer; instead 
of using our inner feeling of duration we read the dial of a 
clock. Thus the observer will generally have no difficulty 
* See, for example, The Nature of the Physical World, pp. 12-16. 


in deciding questions of exactitude. His mutilation will make 
it rather difficult for him to keep general track of what the 
experiment is concerned with; but by the aid of the army of 
assistants that we have allowed him, he will be able to main- 
tain a sufficient watch on all parts of the apparatus. 

I will give one example to show how in scientific practice 
pointer-readings are substituted for diverse sensory data. 
Our ideal observer is supposed to have no sense of the 
graduation of light and shade; therefore when he looks up 
at the night sky all the starry points will look to him alike 
in brilliance. Will not this rather disqualify him as an 
astronomer? Not at all. For let us consider how in practice 
a professional astronomer recognises the differences of bright- 
ness of the stars. It happens that this is the work with which 
my own observatory is now chiefly occupied. We follow 
a method (used also in a few other places) which for the 
brighter stars is found to give by far the most accurate results. 
The light of the star is concentrated by a telescope so as to 
enter a photo-electric cell. But first, how do we know that 
we have got hold of the right star that we can recognise 
again the star which we have been measuring ? Stars are com- 
monly recognised by the patterns that they form with other 
stars crosses, triangles, W's, etc. ; but it will be remembered 
that Einstein has cut down the field of vision of our ideal 
observer so that he cannot see these patterns. No matter. 
The observer at Cambridge would in any case be unable to 
see the patterns, because the telescope is so constructed that 
the observations are made in a closed room without a glimpse 
of the sky; and when the photo-electric apparatus is mounted, 
the observer cannot see through the telescope more than one 
star at a time just as though Einstein had really operated 
on his retina. The star is set for and identified by reading two 
graduated circles attached to the telescope. Thus, even in the 
identification of the star, pointer-readings are substituted for 
other sensory data. 


The light on reaching the photo-electric cell liberates 
electrons from a film of potassium, and these are driven by 
a constant electromotive force (which incidentally is measured 
by another pointer-reading, viz. that of a voltmeter) on to 
the needle of an electrometer. Omitting technical details the 
task of the observer is to watch the pointer-needle of the 
electrometer travel from coincidence with one graduation 
of a scale to coincidence with another graduation, timing it 
with a stop-watch. The stronger the light of the star, the 
faster the passage. So that finally the determination of the 
brightness of the star resolves itself into yet another pointer- 
reading, namely that of the hand of a stop-watch on its 
graduated dial.* 

" One star differeth from another star in glory" wrote the 
apostle. The Nautical Almanac is more precise : 2 Ceti, 4 m *62 ; 
a Andromedae, 2 m -i5; /? Cassiopeiae, 2 m 42; and so on. 
Even the glory of the sun has been systematised in the same 
way as 26 m '7 on the scale of magnitudes. "How art thou 
fallen from heaven, O Lucifer, son of the morning!" All 
thy glory has been turned into the pointer-reading of a 
terrestrial stop-watch. 


If a catalogue of pointer-readings were the ultimate end, we 
might well question whether physical truth were worth the 
seeking. But the pointer-readings are rather the beginning, 
replacing the story teller's romances which from our point 
of view must be looked on as a false start. They constitute 
the material which contains all the recurrencies whereby the 
cryptogram is deciphered, since we find by experience that 
the use of a wider variety of sensory data only leads to 

* The pressing of the stop-watch at the right moment involves a sense 
of touch, so that in this respect the Cambridge observers fall short of our 
theoretical ideal. But the principle would not be affected if an automatic 
method of timing the motion of the needle were substituted. 


redundancies. In later chapters we shall learn some of the 
results of the deciphering; and perhaps you will be persuaded 
that the reconstructed story of the stars is a not inadequate 
compensation for setting aside the familiar story teller's 
romantic imaginations. 

But, it may be suggested, if all observation is reduced to 
coincidences and pointer-readings, can we ever infer from 
it anything but a system of relationship of coincidences and 
pointer-readings? In one sense the answer is No. But if the 
question is put in the form ' * Can we by manipulating pointer- 
readings ever arrive at a knowledge which does not smell 
of pointer-readings?" I suggest that it might equally well 
be asked "Can an artist by manipulating paint ever achieve 
a creation which does not smell of paint ? " But I do not wish 
to set this question aside lightly, for it goes to the very heart 
of the difference between the new and the old scientific 
outlook. We shall see later that a scheme of relationship, or 
a structure, has a significance which can be abstracted from 
the intrinsic nature of that which is the subject of the relation- 
ship. The structure is the object of our search, and when we 
have reached knowledge of the structure we can disregard 
the scaffolding by which we reached it. It does not lessen 
the dignity of the structure that its elements are pointer- 
readings which after all is only the story teller's name for 

If none of the images which constitute our sensory per- 
ception are applicable to the physical world, in what form 
can our knowledge of the physical world be expressed? We 
have deciphered our cryptogram, but the result is a message 
couched in unknown language which we have no hope of 
translating into the language of the story teller. It does not, 
however, follow that it is unintelligible to the mind. 
Perception is only part of our mental outfit, and the language 
of perceiving is only part of the language of knowing. Our 
reading of the cipher of experience leads to an understanding 


of our environment, highly abstract indeed and only to be 
apprehended by the intellect through symbolic expression, 
but nevertheless satisfying to the urge of the human spirit in 
its quest for knowledge. In Chapter xii I will try to explain 
in some detail how a genuine knowledge of the external 
world can be expressed and apprehended without referring 
to perceptual images. Here I will content myself with one 

The sentence which I am now writing can exist in a number 
of forms. It may be a series of sounds perceived by a listener. 
It may be printed in a book. It may be recorded by a gramo- 
phone, and exist as a trace on a disc. It was originally a 
mental composition unuttered and unwritten. There is some- 
thing common to all these forms; and that common element, 
if we can abstract it, constitutes the sentence.* There may 
well be forms of existence of a sentence which are un- 
imaginable to us to-day, just as a hundred years ago it could 
scarcely have been imagined that a sentence could exist as 
a gramophone record. The various forms are described in 
terms of familiar images sounds, discs, black and white 
shapes but the sentence itself is detached from all familiar 
images. (I would again remind you that I am referring to 
the exact words, not to the meaning.) That does not render 
our knowledge of the sentence unsatisfying or incomplete. 
In telling a child of Nelson's famous message to the Fleet, it 
is not necessary to prefix a discourse on the methods of naval 
signalling. And if we could foresee that a hundred years 
hence a certain sentence would pass from one individual to 
another, that would be precise and intelligible knowledge 
of the future, notwithstanding that the transmission might 
be by methods as yet unimagined by us and therefore un- 
specified in terms of familiar images. 

The sentence which constitutes the solution of an ordinary 

* I am not referring to the meaning, which might be conveyed by 
a different sentence or in a different language. 



cryptogram is not associated with any one form of existence. 
Likewise the external world of physics which is the solution 
of the cryptogram of our sensory experience is not associated 
with any one form of existence. This means that when we 
consider experience as a whole, in passing from the mental 
experience to the phenomena of the physical world we do 
not encounter any discontinuity in the form of existence, 
unless we deliberately create a discontinuity.* There is a 
difference, of course for the object of our analysis is to 
differentiate but not a dualism. The older philosophic 
dualism of mind and matter seems to have been that of the 
man who has received one part of his instructions verbally 
and the other part in written form and teels unable to com- 
bine them because of the incompatible nature of sound waves 
and ink. 

By the dropping of redundant sense data we have reduced 
our observational material to pointer-readings, or more 
generally to coincidences. Einstein's general theory of 
relativity (1915) was based on the principle that observable 
data are always describable by coincidences, or, as the 
favourite expression was, " intersections of world-lines". 
Clearly any inference we draw, any structure which we 
ascribe to the external world, must be of such a character 
that it is invariant for any changes which we may make in 
our picture which do not alter these intersections of world- 
lines, i.e. turn intersections into non-intersections. Our 
inference has to have a fluid form. If we conceive a frame- 
work of lines whose intersections correspond to the observed 
coincidences, then however the framework is distorted and 
twisted it will still represent all that we can really know, 
provided that the joints are not tampered with. I suppose 
that a musician who listens to a broadcast performance can 
see in his mind the movements of the fingers and even the 

* Just as we may create a discontinuity of form between a cryptogram 
and its solution by giving one in written form and the other orally. 


swaying of the head and body of the pianist; but setting 
aside his preconceptions all that he can really infer from the 
sound is that certain keys have been struck with greater or 
less force for longer or shorter times, and any scheme of 
movement leading to this result is an equally admissible 
inference. In the same way, setting aside preconceptions, we 
cannot discriminate between the various possible systems of 
structure of the external world which would lead to the same 
sequence of impulses on the extremities of human nerves; 
or since the structure of the nervous system is itself a matter 
of inference, we can transform the whole structure of the 
physical world (nerves included) in any way which does not 
alter the sequence of impulses reaching those points in the 
structure identified as doors of communication with con- 
sciousness. The solution of the cryptogram has (like the 
sentence) many forms of existence, and also (unlike the 
sentence) many equivalent and equally admissible repre- 
sentations within the same form of existence. 

It is this fluidity of representation so different from the 
representation of our environment in the story teller's 
version which first found its way into physics in Einstein's 
theory of relativity. That is why the theory of relativity is 
such an epoch-making breach with tradition. It is interesting 
to notice that this revolution of thought had birth within 
physics itself. I have been arguing that from the very nature 
of our acquaintance with the physical world there must 
necessarily be a fluidity of representation of that which we 
discover about it that many apparently different repre- 
sentations of the world-structure are equivalent in all that 
concerns observation. But it was not by this kind of reasoning 
that the question first arose in physics. The physicist in the 
ordinary course of his work had stumbled upon a multi- 
plicity of representation. He was very much bothered by it. 
He thought it was his duty to decide which representation 
was the "right" one. There were things in Nature which he 



had never doubted were quite definite; the story teller said 
so, and that was good enough for a man who dealt with hard 
facts. Yet Nature by the most artful devices persistently 
refused to disclose anything definite about them. I do not 
mean that Nature is characteristically indefinite and slovenly; 
but she is definite in her own way, not in the story teller's 
way. At last the physicist was forced by his own discoveries 
to consider more philosophically the principles of knowledge 
and the kind of truth that his methods were adapted to 

At present theoretical physics is sharply divided into macro- 
scopic theory and microscopic theory, the former dealing 
with systems on a scale perceptible to our gross senses and 
the latter with the minute atomic substructure underlying 
the gross phenomena. Broadly speaking, macroscopic 
systems are covered by relativity theory and microscopic 
systems by quantum theory. The two theories must ulti- 
mately be amalgamated; the amalgamation is in fact now in 
progress. But for the purposes of a general survey it is easier 
to think of them as distinct. 

Microscopic physics introduces entities molecules, atoms, 
, electrons, protons, photons, etc. which do not appear at all 
in the familiar story. There has been a tendency among 
scientific philosophers to regard these as having a more 
hypothetical status than the objects studied in macroscopic 
physics. Prof. H. Dingle's Science and Human Experience is 
a typical example of this attitude. According to him atoms 
and electrons are unverifiable hypotheses, "existences whose 
unobservability is part of their essential nature" (p. 47). He 
is contrasting them with ordinary "observable" objects, and 
he intends to convey that they have not the same kind of 
connection with human experience as the more ancient 


denizens of the physical world such as sticks and stones and 
stars. This distinction appears to me quite unwarranted. 

An electron is no more (and no less) hypothetical than a 
star. Nowadays we count electrons one by one in a Geiger 
counter, as we count the stars one by one on a photographic 
plate. In what sense can an electron be called more un- 
observable than a star? I am not sure whether I ought to say 
that I have seen an electron; but I have just the same doubt 
whether I have seen a star. If I have seen one, I have seen 
the other. I have seen a small disc of light surrounded by 
diffraction rings which has not the least resemblance to what 
a star is supposed to be; but the name "star" is given to the 
object in the physical world which some hundreds of years 
ago started a chain of causation which has resulted in this 
particular light-pattern. Similarly in a Wilson expansion 
chamber I have seen a trail not in the least resembling what 
an electron is supposed to be; but the name "electron** is 
given to the object in the physical world which has caused 
this trail to appear.* How can it possibly be maintained that 
a hypothesis is introduced in one case and not in the other? 

Thus when we discuss the reality of the physical world 
and the entities which constitute it, we have no reason to 
discriminate between the macroscopic and the microscopic 
entities. It is to be treated as a whole. If the physical world 
is a hypothesis, stars and electrons are hypothetical; if the 
physical world is an inference, stars and electrons are in- 
ferential; if the physical world exists, stars and electrons are 
real. Of course we must not forget that science is progressing, 
and that the various entities now regarded as composing the 
physical world are, as it were, on probation. But this 
domestic uncertainty within the scientific scheme is not here 
a point at issue. It is the principles of physical science rather 
than the interim results which we are examining critically 
in this chapter. Perhaps we may usefully borrow a phrase 

* See Plate i. 


from commerce and finance. The letters, E. & O.E., in a 
document stand, I am told, for "errors and omissions 
excepted". My contention is that atoms, electrons, and 
other entities of microscopic physics (E. & O.E.) are 
hypotheses, inferences or realities according as chairs and 
tables and other commonplace objects of the physical world 
are hypotheses, inferences or realities. 

When we stripped our ideal observer of most of his sense 
organs we left him part of an eye in order that he might 
observe coincidences. Was not this a rather arbitrary selection 
from among his diverse sense organs? Perhaps it was. It was 
enforced entirely by practical considerations; I will not 
defend it on philosophic grounds. I will not enter into an 
argument with my dog as to whether the eye or the nose is 
ideally the more trustworthy organ for exploring the external 
world. All I assert is that in a competition between various 
observers, each allowed only one kind of sensory impression, 
the Einstein observer has up to the present gone furthest in 
discovering the scheme of regularity underlying all sensory 
impressions. The technique of the practical physicist has 
come more and more to depend on observing coincidences 
(pointer-readings and similar measurements). Inferences 
from our other perceptions partially reveal the same scheme 
of regularity in Nature, but they do not go so far in un- 
ravelling it. (The unity of the scheme underlying all our 
diverse perceptions is not an a priori judgment; it is a con- 
clusion, possibly mistaken, which we have drawn from such 
fragments of the scheme as have already been discovered.) 
As the advantage appears to be a purely practical one, I do 
not think we should be justified in attributing a special 
philosophic importance to the perception of coincidences. 
In particular we ought not to displace our mental image of 
coincidence into the external world. 

In the earlier scientific outlook we used to suppose that 
shape and size existed in the external world precisely as they 


appear in our perceptions not like colour which had to be 
represented by a wave-length. Perhaps most physicists would 
now transfer coincidences in the same way, and suppose that 
the coincidences and intersections referred to in the scientific 
description are just like the coincidences and intersections in 
our mental picture. I do not think that this naive displace- 
ment of essentially mental forms of relation is permissible; 
and it is interesting to notice that the quantum theory gives 
a distinct warning against it. 

The observed coincidences of gross matter are, of course, 
only approximate contacts ; but as we deal with smaller and 
smaller particles the conception of coincidence can be refined 
to higher and higher exactitude. If coincidence were the 
key-stone of world structure we should expect to find the 
greatest refinement of it in the theory of atoms and electrons. 
But on the contrary modern physics represents atoms and 
electrons in a scheme which forbids coincidence. There is a 
fundamental law called Pauli's Exclusion Principle which 
asserts that two electrons can never be in the same cell of 
the phase-space in which we represent them. 

In the quantum theory we abandon the last vestige of any 
displacement of the elements of the familiar world into the 
physical world. The connection is not displacement but 
inference. The inferences do not resemble the sense data any 
more than criminals resemble clues. I have hesitated whether 
I ought not to make one reservation. We displace integers 
freely from the familiar world to the physical world. An 
apple in the familiar story has a counterpart in the external 
world; none of our familiar conceptions are appropriate to 
describe the nature of this counterpart, and we can only 
indicate it by a symbol such as X. But at any rate we can 
then say that the counterpart of two apples in the familiar 
story is two X's. I grant that; but I would not like to commit 
myself to the opinion that the twoness of two X's is just like 
the twoness of two apples. In the case of electrons I would 


go further; I do not believe that the twoness of two electrons 
is a bit like the twoness of the two apples in the familiar story. 
In fact multiplicity in the external world should be regarded 
as a property (indescribable in familiar terms) which, being 
by its nature discontinuous, has been correlated to the series 
of arithmetical integers, just as continuous properties are 
correlated to continuous measure numbers. 


In view of the closer contact which now exists between 
science and philosophy, I would like to raise one question 
which affects our cooperation. A feature of science is its 
progressive approach to truth. Is there anything corresponding 
to this in philosophy? Does philosophy recognise and give 
appropriate status to that which is not pure truth but is on the 
way to truth ? Let me here warn the reader that, whilst in this 
opening chapter I set before him the ideal after which we 
strive in sifting the truth about the external world from the 
imaginations of the familiar story teller, I shall not keep to 
the same austere outlook in subsequent chapters. Indeed it 
would lack a sense of proportion to use the steam-hammer 
of critical philosophy to crack every nut on the tree of science. 
It is essential that philosophers should recognise that in 
dealing with the scientific conception of the universe they 
are dealing with a slowly evolving scheme. I do not mean 
simply that they should use it with caution because of its lack 
of finality; my point is that a vehicle of progress is not 
furnished on the same lines as a mansion of residence. The 
scientific aim is necessarily somewhat different from the 
philosophic aim, and I am not willing to concede that it is 
a less worthy aim. 

It would be no aid to science if philosophers enforced on 
us their glimpses of pure truth centuries before our scheme 
was ripe to receive it. Perhaps if it had been guided by 


philosophy, physics would have been relativistic long before 
Einstein; but I feel sure that physics would not now have 
been in so advanced a state if it had never passed through the 
non-relativistic phase of the nineteenth century. So when 
after laborious research physics arrives at "revolutionary 
conclusions*' which philosophy claims to have known from 
its cradle, there are two versions to the story. According to 
one the physicist is a workman of pig-headed disposition 
who would have got on much faster if he had listened to the 
advice of philosophers. The other is that the philosopher is 
an officious spectator who bothers the workman by handing 
him tools before he is ready to use them. I daresay that, as 
is usual in such cases, the truth lies somewhere between the 
two versions. 

I suppose that before concluding I must encounter the 
plain question, Does the external world described in physics 
(E. & O.E.) really exist? But I do not consider it to be a 
"plain question". The difficulty is that the words existence 
and reality require definition like any other terms that we 
employ, and I do not know where to turn for a recognised 
definition. There is no reason to think that a physicist, a 
mathematician and a philosopher attach the same meaning 
to the word "existence". Descartes seems to have believed 
that he existed because he thought. Dr Johnson seems to 
have believed that the stone existed because it was kickable. 
Others have regarded their own existence as a debatable 

For my part, any notion that I have of existing is 
derived from my own existence; so that my own existence 
is a tautological consequence of any definition that I should 
be willing to adopt. Other conscious beings also exist, for 
I am convinced that I must not deny to them the attributes 
I recognise in myself. I thus lay down the rudiments of a 
"web of existence" to which all that enters into knowledge 
is related in various ways. I have tried to show the particular 


way in which the world of physics is related. I expect that 
most people would regard a world related in this way as 
thereby qualified to be considered part of the same web of 
existence; but I cannot feel any great interest in this desire 
to employ a vague instead of an exact description of the 
relation. However, so far as I can judge the meaning of the 
question, the answer appears to be in the affirmative the 
external world described in physics (E. & O.E.) really exists. 
One thing can perhaps usefully be added. I do not think 
chat with any legitimate usage of the word it can be said that 
the external world of physics is the only world that really 


These our actors, 

As I foretold you, were all spirits, and 
Are melted into air, into thin air. 

SHAKESPEARE, The Tempest. 


IT is frequently necessary in the following chapters to refer to 
the chief results of atomic physics and to our general know- 
ledge of atoms, radiation and aether. Many excellent non- 
technical books on the subject are available, and I do not wish 
to linger over a fascinating but oft-told story. It has, how- 
ever, seemed desirable to include here a brief review of our 

We have seen that the ultimate scientific description of the 
physical universe must be divorced from all familiar images; 
but here we shall follow the working conceptions of the 
experimental physicist rather than those of the extreme 
theorist. Scientific conceptions relate to a number of different 
levels, and we do not need to call up the ideas of the pro- 
foundest level for every minor occasion. It would be in- 
appropriate to think in terms of atomic theory in the act of 
stepping off a bus; and similarly the physicist who splits 
atoms may, in a practical sense, quite well understand what 
he is doing without invoking the more recondite conceptions 
of wave mechanics or of the theory of groups. So in this 
chapter I do not at first descend to the foundations, but halt 
at a level which is important because it has supplied a great 
deal of the current vocabulary of physics. My description 
cannot attempt greater accuracy or profundity than that of 
the level to which it belongs. 


It appears that all matter is constructed from two kinds of 
elementary particles called protons and electrons. The proton 
carries a certain definite charge of positive electricity and the 
electron an equal charge of negative electricity. But these 
two kinds of particle are not in all respects the exact opposite 
of one another; for the proton is very much heavier than the 
electron, its mass being about 1847 times as great. 

The true opposite of the electron was discovered about 
two years ago; it is called the positive electron or positron. 
It is a particle of just the same mass as an electron but with 
a positive instead of a negative unit charge. Apparently, 
however, positrons have only a momentary existence. They 
are created during certain kinds of intense discharge of 
energy when cosmic rays fall on matter, or when an atomic 
nucleus is bombarded by fast-moving particles. But after 
travelling a short distance they vanish, having encountered 
ordinary (negative) electrons with the result that mutual 
cancellation takes place. Presumably the proton also has its 
opposite, a negative proton or negatron, but this has not yet 
been discovered. 

Plate I shows the tracks of electrons and positrons, ren- 
dered visible by Prof. C. T. R. Wilson's method which 
causes small drops of water to condense along the tracks. The 
photograph, due to Blackett and Occhialini, shows a shower 
of these particles produced by a single cosmic ray falling on 
copper. A magnetic field was so placed that the electron 
tracks curve to the left and the positron tracks to the right. 
Most of the particles were going too fast to be much de- 
flected, and therefore cannot easily be discriminated; but one 
positron is obvious, and another with smaller curvature is 
fairly evident. 

Both protons land electrons mtufcAie pictured as exceedingly 
small, very much smaller than an apKnu Formerly an electron 
was supposed to have a radius or2- 10^*3 cm. and the proton 
was supposed to be a gre^* A**] mi1W* Kut nm*r we regard 


Blacken and Occhialini 

The tracks pass downwards through a magnetic field which deflects 
electrons to the left, positrons to the right. "One positron track with 
pronounced curvature to the right is easily distinguished. Two electrons 
are seen on the left of the photograph. 


them both as mathematical points. This is not so much a 
correction of the original estimates of size as a recognition 
that the ordinary notions of space break down in the branch 
of physics which deals with these particles. It is found to be 
inappropriate to attribute extension to an electron, though 
we have, as it were, to make it up to the electron in other 

The first step in the construction of matter out of protons 
and electrons is the building of an atom. It is clear that any 
permanent structure must consist of an equal number of 
protons and electrons. For if there were an excess of protons 
there would be a net positive charge ; and this would attract 
any negatively charged electrons in the neighbourhood and 
draw them into the structure until the excess had been 
neutralised. Although the protons and electrons in an atom 
are equal in number, there is great asymmetry in their 
arrangement. All the protons and about half the electrons 
are welded into a structure about io~ u cm. in radius called 
the nucleus; the rest of die electrons, called satellite electrons, 
travel round the nucleus in relatively distant orbits, so that 
the whole atom extends to a radius of about io~ 8 cm. The 
proportion is nearly the same as that of the sun and its 
planetary system; the sun, corresponding to the atomic 
nucleus, has a radius of 430,000 miles, whilst the limits of 
the solar system defined by the orbit of Pluto extend to 
3,600,000,000 miles. We may thus picture an atom as a 
miniature solar system. 

We can specify the different kinds of atoms by giving 
(i) the number of protons in the nucleus, and (2) the number 
of satellite electrons or, what comes to the same thing, the 
excess of the number of protons over the number of electrons 
in the nucleus. The first number gives approximately the 
mass of the atom (taking the mass of a proton as i), for the 
electrons are so light that their masses scarcely count. The 
second numbed Is called the atomic number; it gives the net 


(positive) charge of the nucleus. The chemical name of an 
atom is decided by the atomic number alone. For example, 
an atom with net nuclear charge 17, so that there are 17 
satellite electrons, is called chlorine. But there are two 
common kinds of chlorine, one with 35 protons and 18 
electrons in the nucleus and therefore of atomic weight 35, 
and the other with 37 protons and 20 electrons in the nucleus 
and therefore of atomic weight 37. We speak of these as two 
"isotopes" of chlorine. There are not many phenomena in 
which it makes much difference whether a 35-mass or a 
37-mass chlorine atom is involved. In diffusion experiments 
the former should behave rather more nimbly; but in general 
the differences are so slight that chemists continually worked 
with chlorine for 150 years without discovering that it was 
a mixture of two kinds of atoms. 

The atomic numbers of the elements range from I for 
hydrogen up to 92 for uranium. Elements have been dis- 
covered occupying all but two of these numbers. Many of 
them have, like chlorine, two or more isotopes, so that the 
total number of different kinds of stable atom now known 
exceeds 240. In addition there are many short-lived radio- 
active atoms. It is not certain that 92 is the upper limit for 
an atomic number; in fact, if we admit jerry-built atoms 
which collapse after a few minutes, element No. 93 has 
recently been created artificially by Fermi. 

Element No. I requires special reference. Its simplest form 
is the ordinary hydrogen atom which consists of a proton 
and a satellite electron. It thus differs from all other elements 
in having an elementary particle instead of a complex 
structure for its nucleus. This distinction is so important that 
it is sometimes advantageous to regard matter as being of 
two main varieties, namely hydrogen and not-hydrogen 
(pp. 147, 167). 

Recently an isotope called " heavy hydrogen" has been 
discovered; this has a nucleus consisting of 2 protons and 


I electron, so that it is of atomic weight 2; the net nuclear 
charge is I, and there is just one satellite electron as in ordinary 
hydrogen. By the usual rule the two isotopes would not be 
entitled to separate chemical names; but the circumstances 
are rather exceptional, and heavy hydrogen has been named 
deuterium (or by some writers, diplogen) and given a 
chemical symbol D. Its nucleus, when it occurs without the 
satellite electron, is called a deuton (or diplon). The 
difference between hydrogen and deuterium (of respective 
weights i and 2) is not such a trivial matter as the difference 
between most isotopes; and deuterium and its compounds 
have appreciably different properties from hydrogen and its 
compounds. Naturally the compound D 2 O, or heavy water, 
has received special attention; it is n per cent, heavier than 
ordinary water (H 2 O). A still heavier hydrogen of atomic 
weight 3 has also been discovered; its nuclei (tritons?) consist 
of 3 protons and 2 electrons. 

Another recent discovery is the neutron. This appears to 
be a nucleus consisting of i proton and I electron, so that 
the net charge is zero and there are no satellite electrons. It 
is thus an element of atomic number o. We might describe 
it as an isotope of nothing. From another point of view the 
neutron is a kind of collapsed hydrogen atom; both consist 
of a proton and an electron, the difference being that in the 
neutron they are held close together by nuclear binding and 
in the hydrogen atom more distantly by satellite binding. 
One of the questions we ask ourselves is whether hydrogen 
atoms ever spontaneously collapse into neutrons.* 

Another very familiar particle is the <x particle, f It is the 
nucleus of a helium atom (atomic number 2) consisting of 

* According to some experimenters the mass of a neutron is rather 
greater than that of a hydrogen atom. If so, it contains more energy, 
so that its formation involves an absorption of energy and is not of the 
nature of a spontaneous collapse. 

f A j8 particle is merely a fast-moving electron. 


4. protons and 2 electrons. This appears to be a particularly 
. able combination, and it was formerly thought that within 
the more complex nuclei a large proportion of the protons 
and electrons are grouped as a particles. But later investi- 
gations of the structure of nuclei are adverse to this 
hypothesis. It now appears that each electron is bound to 
a proton so as to form a neutron ; thus the nucleus can be 
treated as an assemblage of neutrons and protons. 

The view is now often advocated that the neutron as a 
simple elementary particle, and that the proton is a complex 
body composed of a neutron and a positron. I do not think 
that this can be accepted as fundamentally true. Doubtless 
there are phenomena for which it is convenient to transpose 
the equation, neutron = pro ton + electron, into proton = neu- 
tron electron, or, since "minus an electron" is equivalent 
to a positron, into proton =neutr on + positron; but I do not 
think the suggestion can be allowed any deeper significance. 


Both the system of satellite electrons and the nucleus itself 
:an be modified or broken by sufficiently energetic dis- 
turbances from outside. It does not require much energy to 
detach the outermost of the satellite electrons ; so that in the 
laboratory, and much more frequently in the stars, we may 
find atoms without their full quota of satellite electrons and 
therefore having a net positive charge. These incomplete 
atoms are called ions. lonisation does not involve any per- 
manent change in the atom any " transmutation of the 
elements" for as soon as the disturbed conditioiis subside 
the nuclear charge, which has been left intact, will attract to 
itself the number of satellite electrons needed to balance it. 

The energy needed to bring about an alteration in the 
nucleus is of a much higher order. But the physicist has at 
his disposal a number of fast missiles electrons^ protons, 


neutrons, deutons, tritons, a particles either projected 
naturally by the discharges of radio-active elements or 
speeded up artificially by applying a large electromotive 
force. Using a sort of machine-gun fire of these missiles, the 
experimenter is able now and then to hit a nucleus with 
sufficient energy for the missile to penetrate and change the 
constitution of the nucleus either by adhesion or disruption. 
The atom is then transmuted into a different element. 

It often happens that the element first created by such a 
bombardment is unstable; so that after a certain short average 
life a rearrangement of the internal structure takes place and 
a particle of some kind is shot out. Thus the original trans- 
mutation is followed by a second spontaneous transmutation. 
The ordinary radio-active atoms, uranium, radium, thorium, 
actinium, are likewise unstable atoms, only they are com- 
paratively long-lived. Their well-known spontaneous trans- 
mutations, which are generally accompanied by the discharge 
of a particles or electrons, are the aftermath of an evolution 
of complex unstable nuclei, which presumably occurred in 
the highly disturbed conditions in the interior of the sun 
some thousands of millions of years ago before the earth 
became separated from the solar mass. 

The mass of a nucleus is not precisely equal to the sum of 
the masses of the protons and electrons composing it; it is 
always a little less. This mass-defect is of great importance 
because it indicates the energy of formation of the nucleus. 
Protons and electrons naturally tend to drop into a configura- 
tion of smallest possible energy; and their tendency to form 
nuclei evidently implies that by so packing themselves their 
total energy is less than when they are apart from one another. 
Thus in the formation of a nucleus energy is set free. 
Actually it is radiated away as high-frequency radiation or 
carried off as kinetic energy by high-speed particles discharged 
during the steps of the formation. It is well known that 
energy and mass are two aspects of the same entity, and when 



the energy departs the corresponding amount of mass also 
departs. Thus the mass-defect records how much energy has 
left the system. 


The chemical, optical and magnetic properties of an atom 
are almost wholly conditioned by the structure of its satellite 
electron system. According to the level of ideas that we are 
now following these electrons describe fixed orbits aboutthe 
nucleus. But it is necessary to insist more strongly than usual 
that what I am putting before you is a model the Bohr model 
atom because later I shall take you to a profounder level of 
representation in which the electron instead of being confined 
to a particular locality is distributed in a sort of probability 
haze all over the atom; and it requires a close study of the 
mathematical equations to see that the two kinds of repre- 
sentation have anything in common. It is doubtless dis- 
concerting to read in one chapter that an electron is confined 
to its groove and cannot pass to another groove without a 
discontinuous jump, and in another chapter that an electron 
in an atom cannot be located anywhere in particular; but 
I suppose that we were once disconcerted to find the world 
in two hemispheres on p. i of an atlas and in Mercator's 
projection on p. 2. 

In Bohr's model there are a limited number of orbits 
available for the electrons. These orbits are laid down by a 
peculiar system of laws given in quantum theory. It is as 
though the field surrounding a nucleus were traversed by a 
number of paths, and electrons roaming in the field were 
instructed to keep to the paths. The orbits are classified in 
groups. Starting from the nucleus there are 2 small circular 
orbits forming group jPC; then come 8 larger orbits (6 circular 
and 2 elliptic) forming group L; then 18 still larger orbits 
forming group M\ and so on. Ideally the series of orbits 
continues up to the limit set by the size of the universe; but 


in practice the territory governed by an atomic nucleus is 
limited by the claims of adjacent nuclei. The larger groups 
of orbits are divided into subgroups corresponding to their 
eccentricities, some being circular and others more or less 
strongly elliptical. 

It is a law that no two electrons may occupy the same 
orbit (Pauli's Exclusion Principle). When the atom is in a 
normal state of quiescence its satellite electrons take up the 
arrangement of minimum energy, which means that in 
general they fill the orbits which are closest to the attracting 
nucleus. But, bearing in mind that the electrons repel one 
another, the problem of finding the arrangement of minimum 
energy is not altogether simple; and when the number of 
satellite electrons is large, it often pays to fill the more 
eccentric orbits of a higher group rather than the circular 
orbits of a lower group. By studying these arrangements it 
has been found possible to explain in detail both the regularity 
and the apparent irregularities in the sequence of chemical 
properties shown in the periodic table of the elements. 

Crudely expressed, the fundamental law of chemistry is 
that a satellite electron likes to belong to a complete group 
or subgroup; it hates to be the odd man out. Helium with 
2 satellite electrons can just complete group K; neon with 
10 satellite electrons can just complete groups K and L. 
Argon (18) completes groups K and L and has 8 electrons 
in group M\ although this does not complete the group, it 
completes the most symmetrical subgroup of group M. 
These atoms are so self-satisfied that they form "inert gases" 
and refuse to enter into combination with other atoms. 
Adding one electron to each of these, we have lithium (3), 
sodium (n) and potassium (19); they accordingly have one 
electron over which must start a new group or subgroup. 
This unhappy electron is called the valency electron, and it is 
responsible for the chemical activity and alkaline nature of 
lithium, sodium and potassium. Taking similarly a step 



backwards, chlorine (17) has 7 electrons in its M group; 
these are as restive and dissatisfied as a party of 7 bridge- 
players. It would be an admirable arrangement for both 
sides if chlorine could borrow sodium's lonely electron to 
complete its group. The arrangement can be made, and the 
two atoms combine to form a molecule of common salt 

Besides matter, which we dissect into protons and electrons, 
the other chief performer in the drama of physics is radiation. 
Radiation is the general name given to electromagnetic 
waves, or waves in the aether which is the continuous back- 
ground between the protons and electrons. These waves may 
be of any length (from crest to crest) or equivalcntly of any 
frequency or pitch. One particular octave can stimulate our 
optic nerves, and within this range of frequency the electro- 
magnetic waves constitute light. Other ranges of frequency 
have other characteristic manifestations. Arranged in order 
of diminishing wave-length and increasing frequency, the 
waves are classified roughly as Hertzian or broadcasting 
waves, infra-red or heat rays, light, ultra-violet or photo- 
graphic rays, X rays, y rays. If the primary cosmic rays are 
electromagnetic waves they are of still higher frequency than 
the y rays, but it now seems more probable that they are 
high-speed particles. 

We shall now consider briefly how atoms and radiation 
interact with one another. When there is energy straying 
round in the form of radiation, or when the atoms are 
jostling one another with energy derived from their high 
temperature, the satellite electrons will not necessarily occupy 
the orbits which correspond to minimum energy. The atom is 
then said to be " excited ". But the atom cannot take up just 
any quantity of energy ; the amount has to be that which will 
lift an electron from one orbit to another vacant orbit. Thus 
for each atom there are a number of characteristic amounts 
of energy which correspond to the different possible transi- 


tions from one orbit to another. These amounts (and no 
others) can be absorbed; or if the atom has already been 
excited these amounts can be emitted in the course of 
returning to the normal state. Here the most characteristic 
rule of quantum theory comes in. When an atom tips out 
a lump of energy into the aether the energy always moulds 
itself into a quantum; that is to say, the energy takes the 
form of a periodic oscillation or wave such that the amount 
of the energy divided by the number of oscillations per 
second is equal to Planck's constant 6-55. io~ 27 erg seconds. 

If you wished to determine the pitch of a bell it would be 
idle to investigate the quantity of energy given out. The two 
measures have no connection, since the same note may be 
struck loudly or softly. But things are different in the 
mechanics of an atom, and the amount of energy emitted 
fixes the pitch or frequency of the resulting radiation. 
Similarly if light is falling on an atom, its frequency deter- 
mines the amount ot energy offered to the atom for ab- 
sorption. Only if the amount coincides with one of the 
possible energies of transition from one orbit to another will 
the atom accept it. Accordingly the series of characteristic 
transition energies of the atom corresponds to a series of 
characteristic frequencies of its radiation. When the radiation 
is examined with a spectroscope and the different frequencies 
are thereby laid out side by side for examination, these 
characteristic frequencies are displayed as the lines of the 

In general the absorption and emission of visual or ultra- 
violet light depends on jumps of the valency electrons in the 
outermost of the occupied orbits. The absorption and 
emission of X rays depends on jumps from and to one of 
the innermost orbits, i.e. in the K or L groups. 

There is another kind of absorption of radiation, called 
photo-electric absorption, in which the electron instead of 
jumping to a higher orbit leaves the atom altogether. This 


naturally requires more energy than the highest orbit-jump, 
and the light must correspondingly be of higher frequency. 
But, provided that it exceeds a certain minimum, no precise 
amount of energy is required; the electron can carry away 
any surplus as kinetic energy of its morion. Absorption of 
this kind accordingly leads to a continuous spectrum which 
begins just about where the line spectrum leaves off. There 
is a corresponding emission of radiation when free electrons 
are captured by ions. 

The quanta of radiation which are tipped out by the atoms 
into the aether in emission, or gathered in by the atoms in 
absorption, are now generally called photons. How far they 
can be said to preserve individual existence between their 
emission by one atom and their absorption by another atom 
is a very obscure question. But at any rate in emission and 
absorption each photon behaves as an indivisible atom of 
radiant energy. Since the amount of energy constituting a 
photon is proportional to the frequency, we must use high- 
frequency radiation (X rays or y rays) if we want a highly 
concentrated packet of radiant energy to let loose anywhere, 
e.g. inside an atom. 


As far as and beyond the remotest stars the world is filled 
with aether. It permeates the interstices of the atoms. Aether 
is everywhere. 

How dense is the aether ? Is it fluid like water or rigid like 
steel? How fast is our earth moving through it? Which way 
do the particles of aether oscillate when an electromagnetic 
wave travels across it? At one time these were regarded as 
among the most urgent questions in physics; but at the end 
of a century's study we have found no answer to any of 
them. We are, however, convinced that the unanswerable- 
ness of these questions is to be reckoned not as ignorance but 
as knowledge. What we have found out is that aether is not 


the sort of thing to which such questions would apply. 
Aether is not a kind of matter. Questions like these could be 
asked about matter but they could not be asked about time, 
for example; and we must reckon aether as one of the 
entities to which they are inappropriate. 

Since aether is not material it has not any of the usual 
characteristics of matter mass, rigidity, etc. but it has 
quite definite properties of its own. We describe the state of 
the aether by symbols, and its characteristic properties by the 
mathematical equations that the symbols obey. 

There is no space without aether, and no aether which does 
not occupy space. Some distinguished physicists maintain 
that modern theories no longer require an aether that the 
aether has been abolished. I think all they mean is that, since 
we never have to do with space and aether separately, we can 
make one word serve for both; and the word they prefer is 
"space". I suppose they consider that the word aether is 
still liable to convey the idea of something material. But 
equally the word space is liable to convey the idea of com- 
plete negation. At all events they agree with us in employing 
an army of mathematical symbols to describe what is going 
on at any point where the aether is or, according to them, 
isn't. "Wheresoever the carcase is, there will the eagles be 
gathered together", and where the symbols of the mathe- 
matical physicist flock, there presumably is some prey for 
them to settle on, which the plain man at least will prefer to 
call by a name suggestive of something more than passive 

Those to whom the word space conveys the idea of 
characterless void are probably more numerous than those 
to whom the word aether conveys the idea of a material jelly ; 
so that aether would seem to be the less objectionable term. 
But it is possible to compromise by using the term "field". 
The field includes both an electromagnetic field and a gravi- 
tational or metrical field; and the army of symbols to which 


I have alluded describes these two fields. Space (in its ordinary 
physical meaning) is the same thing as the metrical field ; for 
the symbols describing the metrical field specify the one 
characteristic that we are accustomed to ascribe to a space, 
viz. its geometry (Euclidean or non-Euclidean). In specifying 
the geometry they specify also the field of gravitation, as 
Einstein showed in his famous theory. We recognise that 
there is an inner unity of the electromagnetic and the metrical 
(gravitational) fields; and the mode of bifurcation of the 
single unified field into these two component fields is, I think, 
fairly well understood.* 

The change in our conception of the world wrought by 
the aether or field theory may be illustrated by an incident 
not infrequent in astronomical observatories. A visitor is 
handed a photograph of some interesting celestial object. 
He is puzzled; he turns it this way and that; but he cannot 
get the hang of the thing. At last the astronomer sees what 
is the trouble "I should have explained. This is a negative. 
The dark markings constitute the object; the bright part is 
only background". The visitor mentally turns the picture 
inside out, and immediately it makes sense. Something like 
a turning inside out of our familiar picture of the world is 
what the aether theory really stands for. Early electrical 
theories focused attention on an electric fluid flowing along 
a wire and treated the space outside the wire as mere back- 
ground. Faraday taught us that, if we would understand the 
phenomena of electricity, the supposed background the 
field outside the wire was the place to attend to. If you can 
make this reversal of the picture, turning space from a 
negative into a positive, so that it is no longer a mere back- 
ground against which the extension and the motion of matter 
is perceived but is as much a performer in the world drama 
as the matter is then you have the gist of the aether theory 
whether you use the word "aether" or not. 

* The Nature of the Physical World, p. 236. 


The reversal of the picture is liable to be carried too far. 
After the great development of the field theory of electro- 
magnetism by Faraday and Maxwell, attention was brought 
back to the more material aspect by the discovery of the 
electron and the development of electron theory by Lorentz 
and Larmor. This reaction in its turn has probably proceeded 
too far, and it would be a gain if the field aspect were more 
emphasised. But by gradually diminishing oscillations we 
are drawing nearer to a unified field-matter theory in which 
neither the field nor the matter is mere background, and one 
is seen to be the necessary complement of the other. 


Hitherto I have not touched the deepest level of ideas in 
physics. Behind the pictures and models which I have been 
describing there is a more profound conception of the 
phenomena, in wliich the electrons and protons are replaced 
by waves. This new form of quantum theory originated in 
a remarkable paper by W. Heisenberg in 1925; the wave 
conception embodied in it is due more especially to L. de 
Broghe and E. Schrodinger. It is usually called Wave 
Mechanics; bur the general term quantum theory must be 
understood to include the new development. 

Let us first understand the relation between the particle in 
the old theory and the wave in the new theory. We have 
seen that the electron (as a particle) has no size ; the conception 
of size does not apply to it. From a geometrical point of 
view it is a point, whose sole characteristic is position. But it 
has also mechanical characteristics, namely momentum and 
energy (or mass) and a more recondite property called " spin ". 
For our purpose it is sufficient to consider position, since 
precisely the same ideas apply to the other characteristics. 

Regarding then position as the sole characteristic, there is 
nothing that we can say about the electron unless we know 


its position. But we may partially know its position ; we may 
know that it is in one or other of two places ; or we may 
know that (owing to the attractions and repulsions) it is more 
likely to be near a proton than near another electron. To 
describe this partial knowledge, let us imagine a fog whose 
density at any place is proportional to the probability that the 
electron is at that place. The mass of fog in any volume then 
represents the probability that the electron is in that volume. 
The fog extends to every corner of the universe where 
(according to our knowledge) there is any possibility that 
the electron may be lurking. If we happen to have exact 
knowledge of the position we can represent it in the same 
picture; the fog is then cleared away from all other parts of 
space and concentrated into a single drop in that position. 

We may identify this "drop" with the electron, that is to 
say it is our pictorial representation of the electron. For we 
have given the name electron to the entity which occupies 
the position, and according to our picture the drop is the 
occupant. But when we go back to partial knowledge a 
distinction appears. In the new picture the drop diffuses into 
fog. In the old picture the electron or drop remains con- 
centrated, but we do not quite know where to represent it. 

Let us, however, continue to study the fog. In the course 
of time the position of the electron changes, and equally the 
positions where it is likely to be are changing. That is to say, 
the distribution of the fog changes. In an actual medium 
changes of density are propagated by waves. That is how 
we come to be concerned with "wave mechanics". Wave 
mechanics examines the laws of propagation of waves 
through our fog, and enables us to calculate how in conse- 
quence the density of the fog changes in different places. We 
can thus trace from time to time where the densest part of 
the fog will be situated. You will remember that the densest 
part represents the place where the electron is most likely 
to be found. Thus wave mechanics achieves essentially the 


same end as ordinary dynamics which traces the motion of 
the electron as a particle; only it does so in a way adapted 
to partial knowledge. It is useful when our data are given in 
the form of probabilities or (what comes to the same thing) 

The waves of fog must not be confused with the electro- 
magnetic or aether waves which constitute radiation. They 
are of an altogether different nature. 

If you have followed me thus far, you will perhaps say 
that I have not really reached a profounder level of ideas. 
I have described an alternative method of treating the pro- 
blem of the movements of electrons, which has turned out 
to be the more powerful method in practice; but I have not 
introduced any real change of conception. As for the fog if 
our knowledge is only partial it is natural that our picture 
should be foggy. I agree. The real change of conception has 
yet to be introduced. 

The crucial point is this. We have discovered the laws of 
propagation of waves in the fog; we have not discovered the 
laws of motion of the electron as a particle. Therefore, what- 
ever be the ultimate truth of things, it is the waves not the 
particles that constitute the world with which the physicist 
of to-day is dealing. 

The older quantum theory which treated the electron as 
a particle succeeded up to a certain point. But it never got 
so far as to formulate a system of laws of motion which 
would cover the jumps of the electron from one orbit to 
another. It was a collection of strange empirical rules rather 
than a systematic theory. No one could foresee what would 
be thq next step in its development what new rule would 
have to be added. Wave mechanics is a much more unified 
theory. All its developments proceed naturally from the 
wave conception, and we do not have to invent ad hoc rules 
as we go along. It is, however, not its aesthetic advantage 
but its practical success that has led to its universal adoption. 


It succeeds better because it attempts less. It does not pretend 
to tell us where the electron is going next; but it docs claim 
to tell us as much about its future position as is actually 
involved in the recurrencies of sensory experience. Errors 
and omissions excepted, wave mechanics enables us to predict 
sensory experience so far as we have any reason to suppose 
that sensory experience is predictable; but it does not predict 
more about the future of the external world than is necessary 
for this special purpose. 

How should we now describe the physical universe or 
"the universe as it is conceived in modern physics"? It is 
difficult to speak consistently. I suppose that we ought to 
mean that conception or formulation which has been 
generally adopted as giving the most complete agreement 
with observation. The formulation assigns certain contents 
and laws, and we are satisfied that by tracing mathematically 
the consequences of these laws we reproduce the diversity 
of phenomena, or more strictly the recurrencies of experience, 
which it is the purpose of physics to analyse; or at least we 
consider that the failures are not such as to cause uneasiness, 
bearing in mind that development of the theory is con- 
tinually proceeding. With that understanding, it cannot be 
said that the content of the universe as it is conceived in 
modern physics consists of a number of particles called 
protons and electrons together with waves of radiation. It is 
no use assigning contents without laws governing them; and 
we have not succeeded in formulating a system of law on 
this basis. In the formulation which must have the credit for 
the most far-reaching success in scientific prediction, the 
content of the universe is the "fog", and the basal laws of 
physics are the laws of propagation of waves of fog the 
wave equations. Now that it has become the actual stuff of 
the universe as it is conceived in physics it is awkward to have 
to refer to it colloquially as fog. I shall sometimes call it "0 " 
that being the symbol by which it enters into our equations, 


though properly speaking $ is a measure of the fog rather 
than the stuff itself. More often I shall call it "probability'*, 
i.e. probability of a particle being present; but that implies 
that there is a universe of particles hovering in the background 
of our thoughts, although we have seen that it cannot 
properly be described as the universe conceived in modern 

We ought therefore to say that on the present view the 
content of the universe consists, not of particles, but of waves 
of 0. But at the same time it must be realised that a universe 
composed of ^ waves necessarily contains a large subjective 
element. Its constituents collect into drops or dissolve into 
fog according as our knowledge of them happens to be 
precise or partial. It is a stage whereon the spirit-actors 
materialise and dissolve as we turn our attention one way 
and another. There is a provision (Heisenberg's Uncertainty 
Principle, p. 97) that as the geometrical characteristics of a 
constituent condense its mechanical characteristics dissolve, 
so that the actor never comes wholly into focus at one and 
the same time. 

We must concede therefore that "the universe as it is con- 
ceived in modern physics" is not identical with what a 
philosopher would call "the objective physical universe". 
When we come to think of it there is 110 reason why it 
should be. The task of physical science is to disclose the 
scheme of the recurrencies in the combined experience of 
conscious beings. We have seen that the universe which con- 
stitutes the solution of this problem must necessarily have 
the characteristics of regularity and externality; we said 
nothing about objectivity. And so it happens that the aim 
of science and the search for an objective universe follow the 
same road up to a certain point and then part company. The 
scientist then has no choice as to which route to follow; he 
can only solve the problem for which our experience pro- 
vides the data. 


Thus in saying that wave mechanics corresponds to a pro- 
founder level of conception I do not mean that it takes us 
closer to the objective world behind the phenomena; I mean 
that it reveals more fully the source of the regularities in our 
experience, which are conditioned as much by our mode of 
acquaintance with the objective world as by the constitution 
of that world. Six years ago* I described wave mechanics as 
"not a physical theory but a dodge and a very good dodge 
too ". If I have changed my view at all, it is in regard to the 
aim of physical theory. If it is still held that the aim of 
physical theory is to describe objective reality, wave mechanics 
is not a physical theory in that sense. 

The nearest we have got to objective reality is the world 
of protons and electrons; that is to say, such a world corre- 
sponds to the level of conception which physics had reached 
before it was forced to deviate towards a different aim. 
Between the universe of our experience and the universe of 
objective reality probability interposes like a smoke screen. 

I will give an example to show that for some purposes an 
atom constructed out of fog (or ^) is a more practical con- 
ception than an atom constructed out of particles. We know 
that the light waves emitted by an atom have a periodicity 
which is characteristic of the atom. It is natural to suppose 
that this periodicity exists within the atom itself and that 
something concerned in the structure of the atom is oscillating 
with that period. In the atom constructed of particles (the 
Bohr model) there is no trace of the period; there is no 
condition or configuration which goes through a cycle in the 
period of the emitted light waves. But in the atom con- 
structed out of </r the period plainly appears; it is the period 
of the "beats" formed by two sets of ^ waves. I do not want 
to overstress the significance of this. I mention it as showing 
that even from a commonsense point of view the change of 
conception is not wholly a change for the worse. As Heisen- 
* The Nature of the Physical World, p. 219. 


berg pointed out, we have to infer the nature of the inside of 
an atom from what we observe coming out of it ; and since the 
most definite tilings coming out of it are certain periodicities 
shown by the spectral lines, the most logical inference would 
seem to be that, whatever else there may be in the atom, 
these periodicities are certainly there.* 


I said earlier that the aether (field, space) has no mass; but it 
would seem that according to a deeper level of conception 
this is not strictly true. By the general relativity theory mass, 
momentum and stress are identified with certain components 
of curvature of space-time or of the metrical field. Now in 
a region where there is no recognised matter or electro- 
magnetic field there is still a certain small natural curvature, 
viz. that specified by the famous "cosmical constant ". The 
mass, momentum and stress equivalent to this curvature 
ought therefore to be ascribed to whatever we suppose to 
occupy such a region, i.e. to the space, field or aether 
whichever term we are using. 

It seems convenient to revive the term aether to express 
the fact that we do not in any region have to deal with 
strictly zero density. This turns out to be a crucial considera- 
tion in connecting relativity theory with quantum theory. 
For the operations of quantum theory (wave mechanics) are 
multiplicative. The theory deals with probabilities which are 
combined by multiplication, not by addition. Now zero is 
a very awkward number to deal with in multiplicative 
operations, and similarly empty space is a very awkward sort 
of abstraction to introduce into quantum theory. The ex- 
istence everywhere of a residual density provided by the 

* I have no high opinion of this argument (for after all the Bohr model 
did not put anything into the atom that had not been observed coming 
out of it) ; but it should appeal to those who stress what are called 
' ' commonsense ideas ' ' . 


natural cosmical curvature thus fits the universe to be the 
field of application of quantum theory. 

We commonly regard completely empty space (devoid 
of mass and of even the most infinitesimal probability of 
containing mass) as being the framework common to both 
theories. Into this empty framework each theory then puts 
its own characteristic entities; the quantum theory inserts a 
probability distribution of electrons and protons, and the 
relativity theory inserts its macroscopically averaged energy 
tensor of matter and electromagnetic fields. But actually 
the conception of an empty framework is foreign to both 
theories, and can indeed only be introduced as a limit. When 
we examine the standard framework which the theories use 
not that which it is commonly imagined they ought to use 
the connection leaps to the eye. In relativity theory the 
norm is, not zero density, but the density corresponding to 
the natural cosmical curvature. In quantum theory the norm 
is, not a region certainly devoid of particles, but one in 
which there is a uniform and iso tropic "a priori probability 
distribution" of the particles and their momenta. The con- 
nection of the two theories lies in the identification of these 
two norms. The mass momentum and stress of the a priori 
probability distribution in quantum theory is the mass 
momentum and stress represented by the natural cosmical 
curvature in relativity theory. We shall deduce important 
consequences from this later. 

Whitehead once said "You cannot have first space and 
then things to put into it, any more than you can have first 
a grin and then a Cheshire cat to fit on to it". To adapt the 
simile to the present state of physics we should have to 
modify it slightly; we should admit the grin provided that 
there were a (non-zero) probability of a cat to fit on to it. 
But leaving aside this minor change the essential truth 
remains. You cannot have space without things or things 
without space; and the adoption of thingless space (vacuum) 


as a standard in most of our current physical thought is a 
definite hindrance to the progress of physics. By this self- 
contradictory and irrelevant conception, we have in our 
current physics made an abstract separation of the theory of 
space (field) from the theory of things (matter) ; and now 
those who are seeking a unified field-matter theory are finding 
it difficult to join them up again. As I have indicated above 
the remedy is to use a norm or standard (common to both 
theories) which does not correspond to complete absence of 

"Nature abhors a vacuum." I think that theoretical 
physics would be wise to follow her example. 



Far better 'tis, to die 

the death that flashes gladness, 

than alone, in frigid dignity, 

to live on high. 

Better, in burning sacrifice, 

be thrown against the world 

to perish, than the sky 

to circle endlessly 

a barren stone. The Shooting Star* 


THE ride of this chapter is ambiguous. It promises a dis- 
cussion of the end of the world, but it does not say which end. 
The world or space-time is four-dimensional and conse- 
quently offers a choice of directions in which we might 
proceed to look for an end; and it is by no means easy to 
describe from a purely physical standpoint the direction in 
which I intend to take you. We shall in fact have to devote 
most of our attention to this preliminary question "Which 


We no longer look for an end of the world in its sjpace 
dimensions. There is reason to believe that, so far as space 
dimensions are concerned, the world is of closed spherical 
type. If weproceed in any direction in space we do not come 
to an enJ7>Tspace, noFdo we continue on and on to infinity; 
bucjlter^avellm a distance, great but not imi^asuxably 
great, we find ourselves^ back at p^r ^tailing pj^jjtaying 
"gone 'round the world". A space that has thi^ re-entrant 
property is said to be finite fa^ unbounded. The~ surface of a 

* Quoted in Nature, Aug. 26, 1933. Author unknown. 


sphere is an example of a finite but unbounded two- 
dimensional space; we have to imagine in the universe the 
same kind of connectivity but with one more dimension. 
I suppose that even if we can to some extent picture such a 
bubble space length, breadth and thickness all lying in the 
film of the bubble it is hard to convince ourselves that the 
picture is not nonsensical as a representation of the space of 
actual experience. But let me remind you that the familiar 
idea of space is the idea of the story teller who lives inside our 
minds. He has never been outside his own doors ; he cannot 
run along to the ends of the nerves and roam into the external 
world to see what it really is that is arousing our sensory 
perceptions. When I say that a finite and unbounded type of 
space is not contradictory to experience, I mean that it is not 
incompatible with the extremities of our nerves being stimu- 
lated by external phenomena in the way requisite to induce 
the actual sequence of our perceptions. So if finite but un- 
bounded space offers the most satisfactory solution of the 
cryptogram, there is no reason why we should not accept 
it as the solution of the cryptogram. 

Spherical space will occupy us in Chapter x and we shall 
not linger over it here. Let us turn to time. The world is 
closed in its three space dimensions, but it is open at botli 
ends in its one time dimension. Proceeding from "here" in 
any spatial direction we ultimately return to "here 1 '; but 
proceeding from "now" towards the future or the past we 
never return to "now". There is no bending round of time 
to bring us back to the momSffweTset out from. In mathe- 
matics we find it convenient to provide for this difference 
between the closed character of space and the open character 
of time by means of the symbol V i; those familiar with 
analytical geometry will recall that the same symboljcrops 
tip iii differentiating between a closed ellipse and an open 

If then we are seeking an end of the world or an infinite 



conrinuarionjor all eternity; wejnust proceed in one ojfthe 
twcTume (Erections. .. How shall we cTedde ^HigEjiCffie.two 
3irecripns^to take ?Jf Imagine yourself in some unfamiliar 
surroundings in space-time billions of miles from Jiejre, 
billions of years from now undergoing experiences that 
you have never undergone before. How would you know 
which were the earlier and which the later events in those 
experiences ? It is said that in a fog an airman sometimes flies 
upside doWn without knowing it. Could one similarly be- 
come inverted in time if none of the accustomed indications 
were discernible ? Or is there everywhere and everywhen in 
the physical universe a signpost with one arm marked "To 
the Future" and the other arm "To the Past"? My first 
business is to hunt for this signpost; for if I mistake the way, 
1 shall lead you to what is no doubt an end of the world, 

but it will be that end which is more usually ^aUed.^|he 

i > 

"""For ordinary purposes the signpost is^ detected Jbjjcon- 
sciousness. Some would perhaps say that consciousness 3oes 
not"Hotlicr about signposts, but wherever it finds itself it 
hurries off on urgent business in some direction, and the 
physicist meekly follows its lead and labels the course that it 
takes "To the Future". It is an important question whether 

- . . ....TT ~r~ *>, - T t ~ "T". 

consciousness in selecting its direction is turned by anything 

.***, .fl-^^^^l'^'^^T^'^V/* ' ' ' " """***?** t- T v **' " *' i <4^*itt~~'~ 

in the physical world. It it is so guided we ought to be able 
to "find me 'particular felhir^nttK^^ 

.,*"* ** *" < " "*"**'""'-"*"'>"" -...** .~-^-- .,..'' "l A 

makes it a one-way street for conscious beings. As scientists 

,!.. * -'"* /..- rr*' ***,... -- ^ - - < 

We are anxious to make the scheme of the physical universe 
as self-contained as possible. We do not want to be dependent 
on consciousness, which is outside the scope of physics, for 
so fundamental a physical distinction as that between past and 
future. If there were nothing apart from our consciousness 
that could discriminate future from past we should have to 
regard the distinction as merely subjective. 
Two rather different questions are involved. Anticipating 


a little, I may say that a signpost for time has been jfouadLitt 
the^pKysical universe, so that we are not wholly dependent 

.^H^faru^'y^*"" 1 'i*">">- >" 'ft* ' ' T ..... rri * " 1 " " "*.,.*- .P-.^YJ 

on tne intuition or consciousness. To^that extent the dis- 
tinction of past and future is objective. But our conscious- 
ness also insists that the distinction is of a particular kind; it 
has a kind of dynamic quality which we can feel though we 
cannot define it. We cannot describe this quality in mathe- 
matical symbols, and we cannot therefore expect the physicist 
to discover it in the external world. Nevertheless ih^d^- 
namicaljjgilire^f time the conception 9f *' becoimng " is 
scTessential a part of our outlook on experience that tde purely 
for distinguishing past and future always 

seems ITveiry inadequate substitute for the going on of time 
wfiiclF we perceive in our consciousness. The statement that 
thiiigs ^become" from past to future seems to convey a 
great deal more than the statement that there exists a way 
of distinguishing past from future. 

The view is sometimes held that the dynamic quality of 
time does not exist in the physical universe and is a wholly 
subjective impression. Experience presen ts the physical world 
as a cinematograph film which is being unrolled in a certain 
direction; but it is suggested that that is a property of the 
way the film is inserted into the cinematograph lantern of 
consciousness, and that there is in the film itself nothing to 
decide which way it should be unrolled.* If this view were 
right the "going on of time' * ought not to appear in our 
picture of the external world. Just as we have dropped the 
old geocentric outlook on the universe, treating it as an 
idiosyncrasy of our own situation as observers, so we should 
drop the dynamic presentation of events the becomingness 
of things treating it as a peculiarity of the process of appre- ! 
hending the world in consciousness. In that case, however, 
we must be careful not to treat the usual past-to-future 

* The two ends are marked distinctively (as we have stated above), 
but that still leaves open the question which is the right mark to begin at. 


presentation of the history of the physical universe as truer 
or more significant than a future-to-past presentation. In 
particular we must drop the theory of evolution, or at least 
set alongside it a theory of anti-evolution as equally true and 
equally significant. 

If anyone holds this view I cannot answer him by argu- 
ment; I can only cast aspersions on his character. If he is a 
professional scientist I say to him: "You are a teacher and 
leader whose duty it is to inculcate a true and balanced out- 
look. But you teach, or without protest allow your colleagues 
to teach, a one-sided doctrine of evolution. You teach it, not 
as a colourless schedule of events, but as if there were some- 
thing significant, perhaps even morally inspiring, in the 
development out of formless chaos of the richness and 
adaptation of our present surroundings, ^hy do you sup- 
press all reference to the sequence from futwfiJtft C&L wjuoi 

L ****^ T -^'^'', ------- T- ,*-, .*H^ ^f rf.MW.MW ^ ** W*ft" * * 

according to you is an equally significant sequence to follow ; 

""-'u , ^ Gf / -'- Jr (I ""**%?, , ,rfuys>f ' *" Jp-'W*" '" i' ^'Jt <-"f*" <f Fl^^W^W"" 

as Kow from the diverse species existing to-day Nature anti- 

evolved clumsier forms, more and more unfitted to survive. 
till she reached the crudity of paleozoic life. Show us how 
from the system of the stars or the planets Nature anti- 
evolved chaotic nebulae. Narrate the whole story of anti- 
progress from future to past, and depict the activity of Nature 
as a force which takes this great work of architecture around 
us and makes a hash of it". 


Setting aside the guidance of consciousness, we discover 
signpost for time in the physical world itself. The signpost i: 
a rather peculiar one, and I would not venture to say that the 
discovery of the signpost amounts to the same thing as th< 
^ "going OH.Q time " in 

But at any rate it provides a unique criterion for7Iiscriminatin 


between past and future, whereas there is no corresponding 
absolute distinction between right and left. The signpost 
depends on a certain measurable physical quantity called 
entropy. Take an isolated system and measure its entropy at 
two instants ti and ti ; the rule is that the instant which corre- 
sponds to the greater entropy is the later. We can thus find 
out by purely physical measurements whether ti is before or 
after h without trusting to the intuitive perception of the 
direction of progress of time in our consciousness. In mathe- 
matical form the rule is that the entropy S fulfils the law 

dSjdt is always positive. 

Thissthe famous Second Law of Thermodynamics. 

be described as a measure 

of tfirwsorganisation of a system. I do not intend that to be 
taken a a definition, because disorganisation is a flexible term 
depending to some extent on our point of view; but in all 
those processes which increase the entropy of a system we 
can see chance creeping in where formerly it was excluded, 
so that conditions which were specialised or systematised 
become chaotic. Many examples can be given of natural 
processes which break up an organised system into a random 
distribution. Plane waves of sunlight all travelling in one 
direction fall on a white sheet of paper and are scattered in 
all directions. The direction of the waves becomes dis- 
organised; accordingly there is an increase of entropy. When 
a solid body moves as a whole, its molecules travel forward 
together; when it is stopped by hitting something, the mole- 
cules begin to move in all directions indiscriminately. It is as 
though the disciplined march of a regiment suddenly stopped, 
and it became a jostling throng of individuals all trying to go 
in different directions. This random motion of the molecules 
is identified with the heat-energy of the body. Quantitatively 
the heat produced by impact is the exact equivalent of the 
lost enerev of motion of the body as a whole, but it has a less 


organised form. Nature keeps strict account of all these little 
wastages of organisation which are continually occurring; 
each is debited against the total stock of organisation con- 
tained in the universe. The balance is always growing less. 
One day it will all be used up. 

Heat, when concentrated, is not fully disorganised energy. 
A further decrease of organisation occurs when the heat 
diffuses evenly so as to bring the body and its surroundings 
to a uniform temperature. In other words heat-energy 
suffers loss of organisation when* it flows from a hotter Bocly 
to Y colder body. This is one of the most common occasions 
of increase j^f entrpp)r(3isofganisation), for unless ''{fie tem- 
perature is everywhere uniform heat is always leaking from 
hotter to colder regions. The fact that a certain amount of 
organisation is retained in a concentrated store of heat enables 
us partially to convert heat into visible motion the reverse 
of what happens at impact. But only partially. To drive a 
train we must put into the engine more heat-energy than 
will appear as energy of motion of the train, the extra quantity 
being needed to make up for its inferior organisation. In that 
way without any creation of organisation we furnish enough 
organised energy to the train; the excess energy, which has 
been drained of organisation as far as practicable, is turned 
out as waste into the condenser of the engine. 

In using entropy as a signpost for time we must be careful 
to treat a properly isolated system. Isolation is necessary 
because a system can gain "organisation by draining itlrom 
other contiguous systems. Evolution ^ shows K ,us_tast. ( more 
highly organised systems develop as time goes on. This may 
Be partly a question of definition, for it does not follow that 
organisation from an evolutionary point of view is to be 
reckoned according to the same measure as organisation from 
the entropy point of view. Butjii^ any case these highly 
developed systems may obtamltheir orjjaiidsaHonBj; a process 
creation* A human beine as he erbws 


from past to future becomes more and more highly organised 
or so he fondly imagines. At first sight this appears to 
contradict the signpost law that the later instant corresponds 
to the greater disorganisation. But to apply the law we must 
make an isolated system of him. If we prevent him from 
acquiring organisation from external sources, if we cut off 
his consumption of food and drink and air, he will ere long 
come to a state which everyone would recognise as a state 
of extreme " disorganisation". 

It is possible for the disorganisation of a system to become 
complete. The state then reached is called thermodynamic 
equilibrium. Entropy can increase no further and, since the 
second law of thermodynamics forbids a decrease, it remains 
constant. Our signpost for time then disappears ; and, so far 
as that system is concerned, time ceases to go on. That does 
not mean that time ceases to exist; it exists and extends just 
as space exists and extends, but there is no longer any 
dynamic quality in it. A state of thermodynamic equilibrium 
is necessarily a state of death, so that no consciousness will be 
present to provide an alternative indicator of " time's arrow ". 

There is no other independent signpost for time in the 
physical world at least no other local signpost; so that if we 
discredit or explain away this property of entropy the dis- 
tinction of past and future disappears altogether. I base this 
statement on a law which has become universally accepted 
in atomic physics, which is called " the Principle of Detailed 


Having found our signpost, let us look around. Ahead there 
is ever-increasing disorganisation in the universe. Although 
the sum total of organisation is diminishing, certain local 
structures exhibit a more and more highly specialised 

* Tlie Nature of the Physical World, p. 79. 


organisation at the expense of the rest; that is the pheno- 
menon of evolution. But ultimately these must be swallowed 
up by the advancing tide of chance and chaos, and the whole 
universe will reach the final state in which there is no more 
organisation to lose. A few years ago we should have said 
that it would end as a uniform featureless mass in thermo- 
dynamic equilibrium; but that does not take into account 
what we have recently learnt as to the expansion of the 
universe. The theory of the expanding universe introduces 
some differences of description but, I think, no essential 
difference of principle, and it will be convenient to consider 
it later in this chapter, adhering for the present to the older 
ideas. When the final heat-death overtakes the universe time 
will extend on and on, presumably to infinity, but there will 
be no definable sense in which it can be said to go on. Con- 
sciousness must have disappeared from the physical world 
before this stage is reached and, dSjdt having vanished, there 
will remain nothing to point the way of progress of time. 

% ~&owleTuno^^ directiontoj^dsthepast. 

Following time backwards we find more anomore oigamA- 
tionindicjgprld. If we are not stopped earlier, we go back 
to a time when the matter and energy of the world had the 
maximum possible organisation. To go back further is im- 
possible. We have come to another end of space-time an 
abrupt end only according to our orientation we call it 
"the beginning". 

I have no philosophical axe to grind in this discussion. 
I am simply stating the results to which our present funda- 
mental conceptions of physical law lead. I am much more 
concerned with the question whether the existing scheme of 
science is built on a foundation firm enough to stand the 
strain of extrapolation throughout all time and all space, than 
with prophecies of the ultimate destiny of material things or 
with arguments for admitting an act of Creation. I find no 


difficulty in accepting the consequences of the present 
scientific theory as regards the future the heat-death of the 
universe. It may be billions of years hence, but slowly and 
inexorably the sands are running out. I feel no instinctive 
shrinking from this conclusion. From a moral standpoint the 
conception of a cyclic universe, continually running down 
and continually rejuvenating itself, seems to me wholly retro- 
grade. Must Sisyphus for ever roll his stone up the hill only 
for it to roll down again every time it approaches the top ? 
That was a description of Hell. If we have any conception 
of progress as a whole reaching deeper than the physical 
symbols of the external world, the way must, it would seem, 
lie in escape from the Wheel of things. It is curious that the 
doctrine of the running-down of the physical universe is so 
often looked upon as pessimistic and contrary to the aspira- 
tions of religion. Since when has the teaching that "heaven 
and earth shall pass away" become ecclesiastically un- 
orthodox ? 

The extrapolation towards the past raises much graver 
difficulty. Philosophically the notion of an abrupt beginning 
of the present order of Nature is repugnant to me, as I think 
it must be to most; and 'even those who would welcome a 
proof of the intervention of a Creator will probably consider 
that a single winding-up at some remote epoch is not really 
the kind of relation between God and his world that brings 
satisfaction to the mind. But I see no escape from our 
dilemma. One cannot say definitely that future develop- 
ments of science will not provide an escape; but it would 
seem that the difficulty arises not so much from a fault in the 
present system of physical law as in the whole relation of the 
method of analysis of experience employed in physical science 
to the actualities with which it deals. The dilemma is this: 
Surveying our surroundings we find them to be far from a 
"fortuitous concourse of atoms". The picture of the world 
as drawn in existing physical theories shows an arrangement 


of the individual atoms and photons which if it originated 
by a chance coincidence would be excessively improbable. 
The odds against it are multillions to i. (I use "multillions" 
as a general term for a number which, if written out in full 
in the usual decimal notation, would fill all the books in a 
large library.) This non-random feature of the world might 
possibly be identified with purpose or design; let us, how- 
ever, non-committally call it anti-chance. We are unwilling 
to admit in physics that there is any anti-chance in the re- 
actions between the billions of atoms and quanta in the 
inorganic systems that we study; and indeed all our experi- 
mental evidence goes to show that these are governed by the 
laws of chance. Accordingly we do not recognise anti-chance 
m tHe laws of physics, but only in the data to which those 
laws are applied. In the corresponding mathematical treat- 
ment we exclude anti-chance from the differential equations 
of physics and relegate it to the boundary conditions for it 
has to be brought in somewhere. One cannot help feeling 
that this segregation of the chance from the anti-chance is a 
characteristic rather of our method of attacking the problem 
than of the objective universe itself. It is as though we ironed 
out a region large enough to include our more immediate 
experience at the cost of puckering in the regions outside. 
We have swept away the anti-chance from the field of our 
current physical problems, but we have not got rid of it. 
When some of us are so misguided as to try to get back 
milliards of years into the past we find the sweepings piled 
up like a high wall, forming a boundary a beginning of 
time which we cannot climb over. 

Without insisting dogmatically on the finality of the second 
law of thermodynamics, we must emphasise that it is very 
deeply rooted in physics. The engineer dealing with the 
practical problems of the heat engine, the quantum physicist 
discussing the laws of radiation, the astronomer investigating 
the interior of a star, the student of cosmic rays tracing 


perhaps the disintegrations of atoms in space beyond the 
galaxy, have all pinned their faith to the rule that the dis- 
organisation or random element can increase but never 
diminish. This faith is not unreasonable when we recall that 
to abandon the second law of thermodynamics means that 
we uproot the signpost of time. 

I have sometimes been taken to task for not sufficiently 
emphasising in my discussions of these problems that the 
laws concerning entropy are a matter of probability, not of 
certainty. I said above that if we observe a system at two 
instants, the instant corresponding to the greater entropy is 
the later. Strictly speaking, I ought to have said that (for a 
smallish system) the chances are, say, io 20 to i that it is the 
later. For by a highly improbable coincidence the multi- 
tudinous particles might at the later instant accidentally 
arrange themselves in a distribution with as much organisa- 
tion as at the earlier instant; just as in shuffling a pack of cards 
there is a possibility that we may accidentally arrange the 
cards in suits or sequences. Some critics seem to have been 
shocked at my lax morality in making the former statement 
when I was well aware of the i in io 30 chance of its being 
wrong. Let me make a confession. I have in the past twenty- 
five years written a number of scientific papers and books, 
broadcasting a good many statements about the physical 
world. I am afraid there are not many of these statements 
for which I can claim that the chance of being wrong is no 
more than i in io*. My average risk is more like I in 10 or 
is that too boastful an estimate ? Certainly if it turns out that 
nine-tenths of what I tell you in this book is correct, I am 
either very fortunate or else very platitudinous. I think that 
if we were not allowed to make statements which had a 
i in io* chance of being untrue, conversation would languish 
somewhat. Presumably the only persons entitled to open 
their lips would be the pure mathematicians. 



One way out of the difficulty of an abrupt beginning has 
sometimes found favour. I oppose it not through any desire 
to retain the present dilemma but because I do not think it 
is a genuine loophole. It depends on the occurrence of chance 
fluctuations. If we have a number of entities moving about 
at random, they will in the course of time go through every 
possible configuration; so that even the most orderly, the 
nlOSt non-chance configuration, will occur by chance if we 
wait long enough 

There once was a brainy baboon 
Who always breathed down a bassoon 

For he said "It appears 

That in billions of years 
I shall certainly hit on a tune". 

When the world has reached complete disorganisation 
(thermodynamic equilibrium) there is still infinite time 
ahead of it, and its elements will have the opportunity to 
take up every possible configuration again and again. If we 
wait long enough a number of atoms will, just by chance, 
arrange themselves as the atoms are now arranged in this 
room; and, just by chance, the same sound waves will come 
from one of the systems of atoms as are now emerging from 
my lips; they will strike other systems of atoms arranged, just 
by chance, to resemble you, and in the same stages of attention 
or somnolence. This mock delivery of the present course of 
Messenger Lectures will repeat itself many times over an 
infinite number of times in fact before t reaches oo. Do 
not ask me whether I really believe, or expect you to believe, 
that this will happen * 

Logic is logic. That's all I say. 
* See p. 68. 


So after the world has reached thermodynamic equilibrium 
the entropy remains steady at its maximum value, except that 
once in a blue moon an absurdly small chance comes off and 
the entropy drops appreciably below its maximum value. 
When this fluctuation has died out there will again be a very 
long wait for another coincidence giving another fluctuation. 
It will take multillions of years, but we have all eternity 
before us. There is no limit to the possible amount of the 
fluctuation; and, if we wait long enough, there will be a 
fluctuation which will take the universe as far from thermo- 
dynamic equilibrium as it is at the present moment. 

The suggestion is that we are now on the down-slope of 
one of these chance fluctuations. Is it an accident that we 
happen to be running down the slope and not toiling up the 
slope ? Not at all. So far as the physical universe is concerned 
the direction of time has been defined to be that in which 
disorganisation increases, so that on whichever slope of the 
mountain we stand the signpost "To the Future" points 
downhill. In fact, on this theory, the going on of time is not 
a property of time in general but of the slope of the fluctuation 
on which we stand. 

We can always argue that anything that can be done by 
arrangement or by specific cause can also be done by chance, 
provided that it is agreed not to count the failures. In this 
case the theory postulates a state of things involving an ex- 
ceedingly rare coincidence, but it also provides an infinite 
time during which the coincidence might (or, it is suggested, 
must) occur. Nevertheless I feel sure that the argument is 

If we put a kettle of water on the fire, there is a chance 
that the water will freeze; for the physical theory of the flow 
of heat indicates that there is very high probability that heat 
will flow from the fire to the kettle but also a trifling chance 
that it will flow the other way. If man goes on putting 
kettles on the fire long enough the chance will one day come 


off, and the individual concerned will be surprised to find a 
lump of ice in his kettle. But it will not happen to me. So 
confident of this am I that even if to-morrow I find ice 
instead of boiling water in the kettle I shall not explain it 
that way. Probably I shall exclaim "The devil's in it". That 
indeed would be a more rational explanation. At present 
I do not believe that devils interfere with cooking arrange- 
ments or other experimental proceedings because I am con- 
vinced by experience that Nature obeys certain uniformities 
which we call laws. I am convinced because these laws have 
been tested over and over again. But it is possible that every 
single observation from the beginning of science has just 
happened to fit in with the laws by a chance coincidence. 
That would, of course, be a highly improbable coincidence, 
but I calculate that it is not quite so improbable as the 
coincidence involved in my kettle of water freezing. So if 
the event happens and I can think of no other explanation, 
I shall have to choose between two highly improbable co- 
incidences: (a) that there is no foundation for the system of 
physical law accepted in science, and that the apparent uni- 
formity of Nature observed up to now is merely a coincidence ; 
(i) that the accepted laws of Nature are true but that I have 
happened upon a phenomenon due to an improbable co- 
incidence. Both explanations do great violence to prob- 
ability, but I think that the former is numerically the less 
unlikely. You will see that when the adverse chance rises to 
multillions a new relation arises between what we commonly 
term the "improbable" and the "impossible". I reckon a 
sufficiently improbable coincidence occurring within the 
supposed laws of Nature as more disastrous than an actual 
violation of the laws ; because my whole reason for accepting 
the laws of Nature rests on the assumption that improbable 
coincidences do not happen or at least that they do not 
happen in my experience. No doubt coincidences described 
as "extremely improbable" occur to all of us, but the im- 


probability is of an utterly different order of magnitude from 
that concerned in the present discussion. 

For that reason if logic assures me that a mock performance 
of these lectures will occur just by fortuitous arrangement of 
atoms sometime before =oo, I would reply that I cannot 
possibly accept that as an explanation of the performance of 
the lectures in = 1934. We must be a little careful over this, 
because there is a trap for the unwary. The crude argument 
is that at a particular epoch (1934) the chance of a fortuitous 
deviation of entropy from its maximum value sufficient to 
admit the phenomenon is too small to be considered 
seriously, and that the fluctuation must therefore be ascribed 
to anti-chance. But the year 1934 is not a random date 
between t= cQ and + 00. We must not argue that be- 
cause fluctuations of the present magnitude occupy only 
i/xth of the time between = oo and t = + oo , therefore the 
chances are x to i against such a fluctuation existing in the 
year 1934. For our present purpose the important charac- 
teristic of the year 1934 is that it is selected as belonging to 
a period during which there exist in the universe beings 
capable of speculating about the universe and its fluctuations. 
It is clear that such creatures could not exist in conditions 
near thermodynamical equilibrium. Therefore it is perfectly 
fair for the supporters of this suggestion to wipe out of the 
calculation all those multillions of years during which the 
fluctuations are less than the minimum required to permit of 
the evolution and existence of mathematical physicists. That 
greatly diminishes x; but the odds are still overpowering. 
The crude assertion would be that (unless we admit some- 
thing which is not chance in the architecture of the universe) 
it is practically certain that the universe will be found to be 
almost in the state of maximum disorganisation. The amended 
assertion is that (unless we admit something which is not 
chance in the architecture of the universe) it is practically 
certain that a universe which contains mathematical physicists 


will be found to be almost in the state of maximum dis- 
organisation which is not inconsistent with the existence of 
such creatures. I think it is clear that neither the crude nor 
the amended version applies. It appears necessary therefore 
to admit anti-chance; and from our present scientific stand- 
point the best we can do with it is to sweep it up into a heap 
at the beginning of time, as I have already described. 

The irreversible dissipation of energy in the universe has been 
a recognised doctrine of science since 1852 when it was for- 
mulated explicitly by Lord Kelvin. Kelvin drew the same 
conclusions about the beginning and end of things as those 
given here except that, since less attention was paid to the 
universe in those days, he considered the earth and the solar 
system. The general ideas have not changed much in eighty 
years; but the recognition of the finitude of space and the 
recent theory of the expanding universe now involve some 
supplementary considerations. 

The conclusion that the total entropy of the universe at 
any instant is greater than at a previous instant dates from a 
time when an "instant" was conceived to be an absolute 
time-partition extending throughout the universe. We have 
to reconsider the matter now that Einstein has abolished these 
absolute instants; but it appears that no change is required. 
I think I am right in saying that it is not necessary that the 
instants should be absolute, or that the time t referred to in 
dS/dt should be a form of absolute time. For the first instant 
we can choose any arbitrary space-like section of space-time 
(smooth or crinkled), and for the second instant any other 
space-like section which does not intersect the first. One of 
these instants will be later throughout than the other;* and 

* That is to say, all observers, whatever their position and motion, 
will encounter them in the same order. 


the total entropy of the universe integrated over the later 
instant will be greater than over the earlier instant. This 
generalisation is made possible by the fact that the energy or 
matter which carries the disorganisation cannot travel from 
place to place faster than light. 

The consequences of introducing die expansion of the 
universe are more difficult to foresee. Fundamental questions 
are raised as to die appropriate way of defming entropy when 
the background conditions are no longer invariaHeTlbelieve 
that the progress of the theory in other directions in the next 
few years will place us in a better position to treat the thermo- 
dynamical problem which it raises, and I prefer not to try to 
anticipate its conclusions. 

Meanwhile it is important to notice that the expansion of 
the universe is another irreversible process. It is a one-way 
characteristic like the increase of disorganisation. Just as the 
entropy of the universe will never return to its present value, 
so the volume of the universe will never return to its present 
value. From the expansion of the universe we reach inde- 
pendently the same outlook as to the beginning and end of 
things that we have here reached by considering the increase 
of entropy. In particular the conclusion seems almost in- 
escapable that there must have been a definite beginning of 
the present order of Nature. The theory of the expanding 
universe adds something new, namely an estimate of the date 
of this beginning. We shall see in Chapter x that from the 
scientific point of view it is uncomfortably recent scarcely 
more than 10,000 million years ago. 

In the expanding universe we can decide which of two 
instants is the later by the criterion that the later instant corre- 
sponds to the larger volume of the universe. (The instants 
are defined as before to be two non-intersecting space-like 
sections of space-time.) This provides an alternative signpost 
for time. But it is only applicable to time taken throughout 
the universe as a whole. The position of entropy as the 



unique local signpost remains unaffected. The fact that the 
direction of time for the universe, regarded as a single system, 
is indicated both by increasing volume and by increasing 
entropy suggests that there is some undiscovered relation 
between the two criteria. That is one of the points on which 
we may expect more light in the next few years. 

By accepting the theory of the expanding universe we are 
relieved of one conclusion which we had felt to be intrinsically 
absurd. It was argued (p. 62) that every possible configura- 
tion of atoms must repeat itself at some distant date. But that 
was on the assumption that the atoms will have only the 
same choice of configurations in the future that they have 
now. In an expanding space any particular congruence be- 
comes more and more improbable. The expansion of the 
universe creates new possibilities of distribution faster than 
the atoms can work through them, and there is no longer any 
likelihood of a particular distribution being repeated. If we 
continue shuffling a pack of cards we are bound sometime 
to bring them into their standard order but not if the 
conditions are that every morning one more card is added 
to the pack. 

So I think after all there will not be a second (accidental) 
delivery of these Messenger Lectures this side of eternity. 


To what extent are conscious beings subject to the second 
law of thermodynamics ? The way in which conscious pur- 
pose might intervene was pointed out by Clerk Maxwell 
who invented a famous "sorting demon". Two adjacent 
vessels contain gas at the same uniform temperature; between 
them there is a very small door. At the door there stands a 
demon. Whenever he sees in the left-hand vessel an unusually 
fast-moving molecule approaching the door, he opens it so 
that the molecule goes through into the right-hand vessel; 


for slow-moving molecules he keeps the door shut and they 
rebound into the left-hand vessel. Similarly he allows slow- 
moving molecules from the right-hand vessel to pass through 
into the left. The result is that he concentrates fast motion in 
the right-hand vessel and slow morion in the left-hand vessel; 
or since the speed of molecular motion corresponds to tem- 
perature, the right-hand vessel becomes hot and the left-hand 
vessel cool. Ideally he might do this without expending any 
energy, since the door might be poised so that an infinitesimal 
effort would open or shut it. But to create a difference of 
temperature of this kind is a gain of organisation; it is the 
opposite of the natural process of disorganisation by the flow 
of heat from a hot to a cold region. Maxwell's demon 
overrides the second law of thermodynamics. 

When in Nature a hot body and a cold body are in contact, 
we find that, as time goes on, the hot body cools and the cold 
body becomes warmer until the temperatures are equalised. 
That is if we have not mistaken the signpost of time. But if 
we happened to have lost our bearings and were viewing 
time backwards, we should see the two bodies first at equal 
temperatures, and then one becoming hotter and the other 
colder precisely the effect that Maxwell's demon achieves. 
Thus effectively the demon reverse^ the signpost of time. 
Being a sorting agent, he is the embodiment of anti-chance; 
and in his domain time appears to run the opposite way from 
that taken in normal systems under the government of chance. 

The mind of man, in virtue of its conscious purpose, must 
play to some extent the part of Maxwell's sorting demon. 
But we must not forget that mind can only make its purposes 
effective in the physical world through its association with a 
body; and whilst the mind may (or may not) be increasing 
organisation the body is always increasing disorganisation. 
It is obvious that (reckoned in physical measure) the organisa- 
tion brought about by our conscious purpose is very small 
compared with that which we consume in eating and 


breathing, so that taken as a whole we do not stem, the 
current of increasing disorganisation. One may hazard the 
suggestion that this is not an accidental limitation, but that 
even the purposive activity of human beings is subject to the 
second law of thermodynamics ; and that the relation of mind 
and body is such that of necessity the amount of organisation 
which the one can put into the world is limited by the amount 
that the other takes out of the world. 

I have sometimes wondered whether it would not be 
possible to baffle Maxwell's sorting demon by one of the 
modern developments of atomic physics, viz. Heisenberg's 
Uncertainty Principle (p. 97). This asserts that a knowledge 
of exact position of a particle is incompatible with a know- 
ledge of exact velocity. I picture the demon scanning the 
approaching molecules for those of large velocity. Since for 
his purpose he has to know their velocities, he must by the 
foregoing principle be uncertain of their positions. He does 
not know how far off they are and how soon they will reach 
the door. So he has to chance the time of opening; and when 
he opens it for the expected high-speed molecule it is quite 
likely that a low-speed molecule will slip through. But I am 
afraid the demon is too clever for me. In some circumstances 
at any rate, his knowledge of both position and velocity, 
though inexact, would be sufficient for the purpose of his 
job; and his mistakes would not be so frequent as to prevent 
a progressive separation of high and low speed molecules. 
Apparently the only way of frustrating the demon is to 
tether him to flesh and blood so that his body spends the 
anti-chance that his mind produces. 

I suppose that to justify the tide of this chapter I ought to 
conclude with a prophecy of what the End of the World 
will be like. It is, of course, not the purpose of our investiga- 
tion to make such prophecies. However, after our serious 
efforts we can perhaps relax. It used to be thought that in 
the end all the matter of the universe would collect into one 


rather dense ball at uniform temperature. But the doctrine 
of spherical space, and more especially the recent results as 
to the expansion of the universe, have changed all that. There 
are unsettled points which prevent a definite conclusion; so 
I will content myself with stating one of several possibilities. 
It has been widely supposed that the ultimate fate of protons 
and electrons is to annihilate one another, and release the 
energy of their constitution in the form of radiation. If so 
it would seem that the universe will finally become a ball of 
radiation, becoming more and more rarified and passing into 
longer and longer wave-lengths. The longest waves of 
radiation are Hertzian waves of the kind used in broad- 
casting. About every 1500 million years this ball of radio 
waves will double its diameter; and it will go on expanding 
in geometrical progression for ever. Perhaps then I may 
describe the end of the physical world as one stupendous 


Thus from the outset we can be quite clear about one very important 
fact, namely, that the validity of the law of causation for the world of 
reality is a question that cannot be decided on grounds of abstract 
reasoning. MAX p IANCKj where is Science Going? p. 113. 

The new theory appears to be well founded on observation, but one 
may ask whether in the future, by development or refinement, it may not 
be made deterministic again. As to this it must be said: It can be shown 
by rigorous mathematics that the accepted formal theory of quantum 
mechanics does not admit of any such extension. If anyone clings to the 
lope that determinism will ever return, he must hold the existing theory 
:o be false in substance; it must be possible to disprove experimentally 
definite assertions of this theory. The determinist should therefore not 
protest but experiment. 

MAX BORN, Naturwissenschaften, 1929, p. 117. 

Whilst the feeling of free-will dominates the life of the spirit, the regularity 
>f sensory phenomena lays down the demand for causality. But in both 
lomains simultaneously the point in question is an idealisation, whose 
latural limitations can be more closely investigated, and which determine 
me another in the sense that the feeling of volition and the demand for 
rausality are equally indispensable in the relation between Subject and 
Dbject which is the kernel of the problem of perception. 

NIELS BOHR, Naturwissenschaften, 1930, p. 77. 

fl7e must await the further development of science, perhaps for centuries, 
>efore we can design a true and detailed picture of the interwoven 
exture of Matter, Life and Soul. But the old classical determinism of 
lobbes and Laplace need not oppress us any longer. 

HERMANN WEYL, The Open World, p. 55. 


r^t^yetTS'-agOYi'^ticHSy every physicist o*epte was-, or 
>elieve^f himself to be, <a determinist, at any rate so far as 
norganic phenomena are concerned. He believed he had 
:ome across a scheme of strict causality regulating the 


sequence of phenomena. It was considered to be the primary 
aim of science to fit as much of the universe as possible into 
such a scheme; so that, as a working belief if not as a philo- 
sophical conviction, the causal scheme was always held to 
be applicable in default of overwhelming evidence to the 
contrary. In fact, the methods, definitions and conceptions 
of physical science were so much bound up with the 
hypothesis of strict causality that the limits (if any) of the 
scheme of causal law were looked upon as the ultimate limits 
of physical^ science.^ No serious doubt was entertained that 
tliis aeterminisin covered all inorganic phenomena. How 
far it applied to living or conscious matter or to conscious- 
ness itself was a matter of individual opinion ; but there was 
naturally a reluctance to accept any restriction of an outlook 
which had proved so successful over a wide domain. 

Then rather suddenly determinism faded out of theoretical 
physics. Its exit has been received in various ways. Some 
writers are incredulous and cannot be persuaded that deter- 
minism has really been eliminated from the present founda- 
tions of physical theory. Some think that it is no more than 
a domestic change in physics, having no reactions on general 
philosophic thought. Some decide cynically to wait and see 
if determinism fades in again. 

The rejection of determinism is in no sense an abdication 
of scientific method. It is rather the fruition of a scientific 
method which had grown up under the shelter of the old 
causal method and has now been found to have a wider 
range. It has greatly increased the power and precision^ of 
the mathematical theory of Observed phenomena. On the 
other hand I cannot agree with those who belittle the 
philosophical significance of the change. The withdrawal of 
physical science from an attitude it had adopted consistently 
for more than 200 years is not to be treated lightly; and it 
provokes a reconsideration of our views as to one of the 
most perplexing problems of our existence. 


In a subject which arouses so much controversy it seems 
well to make clear at the outset certain facts regarding the 
extent of the change as to which there has frequently been 
a misunderstanding. Firstly, it is not suggested that deter- 
minism has been disproved. What we assert is that physical 
science is no longer based on determinism. Is it difficult to 
grasp this distinction? If I were asked whether astronomy 
has disproved the doctrine that "the moon is made of green 
cheese" I might have some difficulty in finding really con- 
clusive evidence; but I could say unhesitatingly that the 
doctrine is not the basis of present-day selenography. 
Secondly, the denial of determinism, or as it is often called 
"the law of causality", does not mean that it is denied that 
effects may proceed from causes. The common regular 
association of cause and effect is a matter of experience; the 
law of causality is an extreme generalisation suggested by 
this experience. Such generalisations are always risky. To 
suppose that in doubting the generalisation we are denying 
the experience is like supposing that a person who doubts 
Newton's (or Einstein's) law of gravitation denies that apples 
fall to the ground. The first criterion applied to any theory, 
deterministic or indetefnliftistic, is that it must accoJIS^for 
the regularities in our sensory experience notably our ex- 
perience that certain effects regularly Follow certain causes. 
Thirdly, the admission of indeterminism in the physical 
universe does not immediately clear up all the difficulties 
iot even all the physical difficulties connected with Free 
Will. But it so far modifies the problem that the door is not 
marred and bolted for a solution less repugnant to our deepest 
ntuitions than that which has hitherto seemed to be forced 
apon us. 

Let us be sure that we agree as to what is meant by 
ietergginimi^ I quote three definitions or descriptions for 
four ccmsi3eration. The first is by a mathematician (Laplace) : 

We ought then to regard the present state of the universe as 


the effect of its antecedent state and the cause of the state that is 
to follow. An intelligence, who for a given instant should be 
acquainted with all the forces by which Nature is animated and 
with the several positions of the entities composing it, if further 
his intellect were vast enough to submit those data to analysis, 
would include in one and the same formula the movements of 
the largest bodies in the universe and those of the lightest atom. 
Nothing would be uncertain for him; the future as well as the 
past would be present to his eyes. The human mind in the per- 
fection it has been able to give to astronomy affords a feeble out- 
line of such an intelligence All its efforts in the search for truth 
tend to approximate without limit to the intelligence we have 
just imagined. 

The second is by a philosopher (C. D. Broad) : 

"Determinism" is the name given to the following doctrine. 
Let S be any substance, iff any characteristic, and t any moment. 
Suppose that 5 is in fact in the state a with respect to ^ at t. Then 
the compound supposition that every tiling else in the world should 
have been exactly as it in fact was, and that S should instead have 
been in one of the other two alternative states with respect to t/t 
is an impossible one. [The three alternative states (of which a is 
one) are: to have the characteristic 0, not to have it, and to be 

The third is by a poet (Omar Khayyam) : 

With Earth's first Clay They did the Last Man's knead, 
And then of the Last Harvest sow'd the Seed: 
Yea, the first Morning of Creation wrote 
What the Last Dawn of Reckoning shall read. 

I regard the poet's definition as my standard. There is no 
doubt that his words express what is in our minds when we 
refer to determinism.. In saying that the physical universe as 
now pictured is not a universe in which "the first morning 
of creation wrote what the last dawn of reckoning shall 
read", we make it clear that the abandonment of deter- 
minism is no technical quibble but is a fundamental change 


of outlook. The other two definitions need to be scrutinised 
suspiciously; we are afraid there may be a catch in them. In 
fact I think there is a catch in them.* 

It is important to notice that all three definitions introduce 
the time element. Determinism postulates not merely causes 
but pre-existing causes. Determinism means predetermina- 
tion. Hence in any argument about determinism the dating 
of the alleged causes is all-important; we must challenge 
them to produce their birth-certificates. 

In the passage quoted from Laplace a definite aim of science 
is laid down. Its efforts "tend to approximate without limit 
to the intelligence we have just imagined ", i.e. an intelligence 
who from the present state of the universe could foresee the 
whole of future progress down to the lightest atom. This aim 
was accepted without question until recent times. But the 
practical development of science is not always in a direct line 
with its ultimate aims ; and about the middle of the nineteenth 
century there arose a branch of physics (thermodynamics) 
which struck out in a new direction. Whilst striving to 
perfect a system of law that would predict what certainly will 
happen, physicists also became interested in a system which 
predicts what probably will happen. Alongside the super- 
intelligence imagined by Laplace for whom "nothing would 
be uncertain" was placed an intelligence for whom nothing 
would be certain but some things would be exceedingly 
probable. If we could say of this latter being that for him 
all the events of the future were known with exceedingly 
high probability, it would be mere pedantry to distinguish 
him from Laplace's being who is supppsed to know them 
with certainty. Actually, however, the new being is supposed 

* The catch that I suspect in Broad's definition is that it seems to convey 
no meaning without further elucidation of what is meant by the sup- 
position being an impossible one. He does not mean impossible because 
it involves a logical contradiction. The supposition is not rejected as being 
contrary to logic nor as contrary to fact, But for a third reason undefined. 


to have glimpses of the future of varying degrees of proba- 
bility ranging from practical certainty to entire indefiniteness 
according to his particular field of study. Generally speaking 
his predictions never approach certainty unless they refer to 
an average of a very large number of individual entities. 
Thus the aim of science to approximate to this latter in- 
telligence is by no means equivalent to Laplace's aim. I shall 
call the aim defined by Laplace the primary aim, and the new 
aim introduced in the science of thermodynamics the 
secondary aim. 

We must realise that the two aims are distinct. The pre- 
diction of what will probably occur is not a half-way stage 
in the prediction of what will certainly occur. We often 
solve a problem approximately, and subsequently proceed 
to second and third approximations, perhaps finally reaching 
an exact solution. But here the probable prediction is an end 
in itself; it is not an approximate attempt at a certain 
prediction. The methods differ fundamentally, just as the 
method of diagnosis of a doctor who tells you that you have 
just three weeks to live differs from that of a Life Insurance 
Office which tells you that your expectation of life is 18-7 
years. We can, of course, occupy ourselves with the secondary 
aim without giving up the primary aim as an ultimate goal; 
but a survey of the present state of progress of the two aims 
produces a startling revelation. 

The formulae given in modern textbooks on quantum 
theory which are continually being tested by experiment 
and used to open out new fields of investigation are ex- 
clusively concerned with probabilities and averages. This is 
quite explicit. The "unknown quantity" which is chased 
from formula to formula is a probability or averaging factor. 
The quantum theory therefore contributes to the secondary 
aim, but adds nothing to the primary Laplacian aim which 
is concerned with causal certainty. But further it is now 
recognised that the classical laws of mechanics and electro- 


magnetism (including the modifications introduced by re- 
lativity theory) are simply the limiting form assumed by the 
formulae of quantum theory when the number of individual 
quanta or particles concerned is very large. This connection 
is known as Bohr's Correspondence Principle. The classical 
laws are not a fresh set of laws, but are a particular adaptation 
of the quantum laws. So they also arise from the secondary 
scheme. We have already mentioned that it is when a very 
large number of individuals are concerned that the pre- 
dictions of the secondary scheme have a high probability 
approaching certainty. Consequently the domain of the 
classical laws is just that part of the whole domain of secondary 
law in which the probability is so high as to be practically 
equivalent to certainty. That is how they came to be mistaken 
for causal laws whose operation is definitely certain. Now 
that their statistical character is recognised they are lost to the 
primary scheme. When Laplace put forward his ideal of a 
completely deterministic scheme he thought he already had 
the nucleus of such a scheme in the laws of mechanics and 
astronomy. That nucleus has now been transferred to the 
secondary scheme. Nothing is left of the old scheme of 
causal law, and we have not yet found the beginnings of a 
new one. 

Measured by advance towards the secondary aim, the 
progress of science has been amazingly rapid. Measured by 
advance towards Laplace's aim its progress is just nil. 

Laplace's aim has lapsed into the position of other former 
aims of science the discovery of the elixir of life, the philo- 
sopher's stone, the North- West Passage aims which were 
a fruitful inspiration in their time. We are like navigators on 
whom at last it has dawned that there are other enterprises 
worth pursuing besides finding the North- West Passage. 
I need hardly say that there are some old mariners who regard 
these new enterprises as a temporary diversion and predict 
an early return to the "true aim of geographical exploration ". 



Let us examine how the new aim of physics originated. We 
observe certain regularities in the course of phenomena and 
formulate these as laws of Nature. Laws can be stated 
positively or negatively, "Thou shah" or "Thou shalt not". 
For the present purpose we shall formulate them negatively. 
Here are two regularities in the sensory experience of most 
of us: 

(a) We never come across equilateral triangles whose 
angles are unequal. 

(b) We never come across thirteen hearts in a hand dealt 
to us at Bridge. 

In our ordinary outlook we explain these regularities in 
fundamentally different ways. We say that the first holds 
because a contrary experience is impossible ; the second because 
a contrary experience is too improbable. 

This distinction is theoretical. There is nothing in the 
observations themselves to suggest to which type a particular 
observed regularity belongs. We recognise that "impossible" 
and "too improbable" are both adequate explanations of any 
observed uniformity of experience; and formerly physics 
rather haphazardly explained some uniformities one way and 
others the other way. But now the whole of physical law 
(so far discovered) is found to be comprised in the secondary 
scheme which deals only with probabilities; and the only 
reason assigned for any regularity is that the contrary is too 
improbable. Our failure to find equilateral triangles with 
unequal angles is because such triangles are too improbable. 
Of course, I am not here referring to the theorem of pure 
geometry; I am speaking of a regularity of sensory experience 
and refer therefore to whatever measurement is supposed to 
confirm this property of equilateral triangles as being true 
of actual experience. Our measurements regularly confirm 
it to the highest accuracy attainable, and no doubt will always 


do so; but according to the present physical theory that is 
because a failure could only occur as the result of an extremely 
unlikely coincidence in the behaviour of the vast number of 
particles concerned in the apparatus of measurement. 

The older view, as I have said, recognised two types of 
natural law. The earth keeps revolving round the sun because 
it is impossible that it should run away. That is the primary 
or deterministic type. Heat flows from a hot body to a cold 
body because it is too improbable that it should flow the 
other way. That is the secondary or statistical type. On the 
modern theory both regularities belong to the statistical 
type it is too improbable that the earth should run away 
from the sun.* 

So long as the aim of physics is to bring to light a deter- 
ministic scheme, the pursuit of secondary law is a blind alley 
since it leads only to probabilities. The determinist is not 
content with a law which ordains that, given reasonable luck, 
the fire will warm me; he agrees that that is the probable 
event, but adds that somewhere at the base of physics there 
are other laws which ordain just what the fire will do to me, 
luck or no luck. 

To borrow an analogy from genetics, determinism is a 
dominant character. Granting a system of primary law, we 
can (and indeed must) have secondary ^deterministic laws 
derivable from it stating what will probably happen under 
that system. So for a long time determinism watched with 
equanimity the development within itself of a subsidiary 
indeterministic system of law. What matter? Deterministic 
law remains dominant. It was not foreseen that the child 
would grow to supplant its parent. There is a game called 
"Think of a number". After doubling, adding, and other 

* "Impossible** therefore disappears from our vocabulary except in 
the sense of involving a logical contradiction. But the logical contra- 
diction or impossibility is in the description, not in the phenomenon 
which it attempts but (on account of the contradiction) fails to describe. 


calculations, there comes the direction "Take away the 
number you first thought of". Determinism is now in the 
position of the number we first thought of. 

The growth of secondary law whilst still under the 
dominant deterministic scheme was remarkable, and whole 
sections of physics were transferred to it. There came a time 
when in the most progressive branches of physics it was used 
exclusively. The physicist might continue to profess allegiance 
to primary law but he ceased to use it. Primary law was the 
gold stored in the vaults; secondary law was the paper 
currency actually used. But everyone still adhered to the 
traditional view that paper currency needs to be backed by 
gold. As physics progressed the occasions when the gold 
was actually produced became rarer until they ceased alto- 
gether. Then it occurred to some of us to question whether 
there still was a hoard of gold in the vaults or whether its 
existence was a mythical tradition. The dramatic ending of 
the story would be that the vaults were opened and found 
to be empty. The actual ending is not quite so simple. It 
turns out that the key has been lost, and no one can say for 
certain whether there is any gold in the vaults or not. But 
I think it is clear that, with either termination, present-day 
physics is off the gold standard. 


The nature of the indeterminism now admitted in the 
physical world will be considered in more detail in the next 
chapter. I will here content myself with an example showing 
its order of magnitude. Laplace's ideal intelligence could 
foresee the future positions of objects from the heaviest bodies 
to the lightest atoms. Let us then consider the lightest particle 
we know, viz. the electron. Suppose that an electron is given 
a clear course (so that it is not deflected by any unforeseen 


collisions) and that we know all that can be known about it 
at the present instant. How closely can we foretell its position 
one second later? The answer is that (in the most favourable 
circumstances) we can predict its position to within about 
1 1 inches not closer. That is the nearest we can approximate 
to Laplace's super-intelligence. The error is not large if we 
recall that during the second covered by our prediction the 
electron may have travelled 10,000 miles or more. 

The uncertainty would, however, be serious if we had to 
calculate whether the electron would hit or miss a small 
target such as an atomic nucleus. To quote Prof. Born: 
"If Gessler had ordered William Tell to shoot a hydrogen 
atom off his son's head by means of an a particle and had 
given him the best laboratory instruments in the world 
instead of a cross-bow, Tell's skill would have availed him 
nothing. Hit or miss would have been a matter of chance". 

For contrast take a mass of -ooi milligram which must 
be nearly the smallest mass handled microscopically. The 
indeterminacy is much smaller because the mass is larger. 
Under similar conditions we could predict the position of 
this mass a thousand years hence to within 5^5 of a milli- 

This indicates how the indeterminism which affects the 
minutest constituents of matter becomes insignificant in 
ordinary mechanical problems, although there is no change 
in the basis of the laws. It may not at first be apparent that 
the indeterminacy of if inches in the position of the electron 
after the lapse of a second is of any great practical importance 
either. It would not often be important for an electron pur- 
suing a straight course through empty space; but the same 
indeterminism occurs whatever the electron is doing. If it 
is pursuing an orbit in an atom, long before the second has 
expired the indeterminacy amounts to atomic dimensions; 
that is to say, we have altogether lost track of the electron's 
position in the atom. Anything which depends on the 


relative location of electrons in an atom is unpredictable 
more than a minute fraction of a second ahead. 

For this reason the break-down of an atomic nucleus, such 
as occurs in radio-activity, is not predetermined by anything 
in the existing scheme of physics. All that the most complete 
theory can prescribe is how frequently configurations 
favouring an explosion will occur on the average; the in- 
dividual occurrences of such a configuration are unpredictable. 
In the solar system we can predict fairly accurately how 
many eclipses of the sun (i.e. how many recurrences of a 
special configuration of the earth, sun and moon) will 
happen in a thousand years ; or we can predict fairly accurately 
the date and time of each particular eclipse. The theory of 
the second type of prediction is not an elaboration of the 
theory of the first; the occurrence of individual eclipses 
depends on celestial mechanics, whereas the frequency of 
eclipses is purely a problem of geometry. In the atom, which 
we have compared (p. 29) to a miniature solar system, there 
is nothing corresponding to celestial mechanics or rather 
mechanics is stifled at birth by the magnitude of the in- 
determinacy but the geometrical theory of frequency of 
configurations remains analogous. 

The future is never entir^ly^.deiermined by the past, norj 
is it ever entirely jdetached^^Efi .have referred jco several! 
phenomena in which the future **jw($i<&lLy... determined * t -the' 
break-down of a radium nucleus is an example of a pheno- 


menon in which the future is ^practi 

But, you will say, the fact that physics assigns no charac- 
teristic to the radium nucleus predetermining the date at 
which it will break up, only means that that characteristic 
has not yet been discovered. You readily agree that we cannot 
predict the future in all cases ; J^jjgby Jalame Nature^rather 
thajp, Qtt own ignorance? lifthe ra3mnT^tom were an 
exception, it woulcl be natural to suppose that there is a 

......... """ 6-2 


determining characteristic which, when it is found, will bring 
it into line with other phenomena. But the radio-active 
atom was not brought forward as an exception; I have 
mentioned it as an extreme example of that which applies 
in greater or lesser degree to all kinds of phenomena. There 
is a difference between explaining away an exception and 
explaining away a rule. 

The persistent critic continues, "You are evading the 
point. I contend that there are characteristics unknown to 
you which completely predetermine not only the time of 
break-up of the radio-active atom but all physical phenomena. 
How do you know that there are not? You are not om- 
niscient?" I can understand the casual reader raising this 
question; but when a man of scientific training asks it, he 
wants shaking up and waking. Let us try the effect of a 

About the year 2000, the famous archaeologist Prof. 
Lambda discovered an ancient Greek inscription which 
recorded that a foreign prince, whose name was given as 
Kocv8e{KAr)$, came with his followers into Greece and estab- 
lished his tribe there. The Professor anxious to identify the 
prince, after exhausting other sources of information, began 
to look through the letters C and K in the Encyclopaedia 
Athenica. His attention was attracted by an article on 
Canticles who it appeared was the son of Solomon. Clearly 
that was the required identification; no one could doubt that 
KavSeiKAris was the Jewish Prince Canticles. His theory 
attained great notoriety. At that time the Great Powers of 
Greece and Palestine were concluding an Entente and the 
Greek Prime Minister in an eloquent peroration made 
touching reference to the newly discovered historical ties of 
kinship between the two nations. Some time later Prof. 
Lambda happened to refer to the article again and discovered 
that he had made an unfortunate mistake; he had misread 
"Son of Solomon" for "Song of Solomon". The correction 


was published widely, and it might have been supposed that 
the Canticles theory would die a natural death. But no; 
Greeks and Palestinians continued to believe in their kinship, 
and the Greek Minister continued to make perorations. 
Prof. Lambda one day ventured to remonstrate with him. 
The Minister turned on him severely, "How do you know 
that Solomon had not a son called Canticles? You are not 
omniscient". The Professor, having reflected on the rather 
extensive character of Solomon's matrimonial establishment, 
found it difficult to reply. 

The curious thing is that the determinist who takes this 
line is under the illusion that he is adopting a more modest 
attitude in regard to our scientific knowledge than the 
indeterminist. The indeterminist is accused of claiming 
omniscience. I do not bring quite the same countercharge 
against the determinist; but surely it is only the man who 
thioJks himself nearly omniscient who would have the 
audacity to enumerate the possibilities which (it occurs to 
hini) might exist unknown to him. I suspect that some of the 
other chapters in this book will be criticised for including 
hypotheses and deductions for which the evidence is con- 
sidered to be insufficiently conclusive; that is inevitable if 
one is to give a picture of physical science in the process of 
development and discuss the current problems which occupy 
our thoughts. I tremble to think what the critics would say 
if I included a conjecture solely on the ground that, not being 
omniscient, I do not know that it is false. 

I have already said that determinism is not disproved by 
physics. But it is the determinist who puts forward a positive 
proposal and the onus of proof is on him. He wishes to base 
on our ordinary experience of the sequence of cause and 
effect a wide generalisation called the Principle of Causality. 
Since physics to-day represents this experience as the result 
of statistical laws without any reference to the principle of 
causality, it is obvious that the generalisation has nothing to 


commend it so far as observational evidence is concerned. 
The indeterminists therefore regard it as they do any other 
entirely unsupported hypothesis. It is part of the tactics of 
the advocate of determinism to turn our unbelief in his 
conjecture into a positive conjecture of our own a sort of 
Principle of Uncausality. The indetermmist is sometimes said 
to postulate "something like free-will" in the individual 
atoms. Something tike is conveniently vague; the various 
mechanisms used in daily life have their obstinate moods and 
may be said to display something like free-will. But if it is 
suggested that we postulate psychological characters in the 
individual atoms of the kind which appear in our minds as 
human free-will, I deny this altogether. We do not discard 
one rash generalisation only to fall into another equally rash. 


When determinism was believed to prevail in the physical 
world, the question naturally arose, how far did it govern 
human activities ? The question has often been confused by 
assuming that human activity belongs to a totally separate 
sphere a mental sphere. But man has a body as well as a 
mind. The movements of his limbs, the sound waves which 
issue from his lips, the twinkle in his eye, are all phenomena 
of the physical world, and unless expressly excluded would 
be predetermined along with other physical phenomena. We 
can, if we like, distinguish two forms of determinism: 
(i) The scheme of causal law predetermines all human 
thoughts, emotions and volitions; (2) it predetermines human 
actions but not human motives and volitions. The second 
seems less drastic and probably commends itself to the liberal- 
minded, but the concession really amounts to very little. 
Under it a man can think what he likes, but he can only say 
that which the laws of physics preordain. 
The essential point is that, if determinism is to have anv 


definable meaning, the domain of deterministic law must be 
a closed system; that is to say, all the data used in predicting 
must themselves be capable of being predicted. Whatever 
predetermines the future must itself be predetermined by the 
past. The movements of human bodies are part of the com- 
plete data of prediction of future states of the material 
universe; and if we include them for this purpose we must 
include them also as data which (it is asserted) can be 

We must also note a semi-deterministic view, which 
asserts determinism for inorganic phenomena but supposes 
that it can be overridden by the interference of consciousness. 
Determinism in the material universe then applies only to 
phenomena in which it is assured that consciousness is not 
intervening directly or indirectly. It would be difficult to 
accept such a view nowadays. I suppose that most of those 
who expect determinism ultimately to reappear in physics 
do so from the feeling that there is some kind of logical 
necessity for it; but it can scarcely be a logical necessity if it 
is capable of being overridden. The hypothesis puts the 
scientific investigator in the position of being afraid to prove 
too much; he must show that effect is firmly linked to cause, 
but not so firmly that consciousness is unable to break the 
link. Finally we have to remember that physical law is 
arrived at from the analysis of conscious experience; it is the 
solution of the cryptogram contained in the story of con- 
sciousness. How then can we represent consciousness as being 
not only outside it but inimical to it? 

The revolution of theory which has expelled determinism 
from present-day physics has therefore die important con- 
sequence that it is no longer necessary to suppose that human 
actions are completely predetermined. Although the door 
of human freedom is opened, it is not flung wide open; only 
a chink of daylight appears. But I think this is sufficient to 
justify a reorientation of our attitude to the problem. If our 


new-found freedom is like that of the mass of 'OOi mgm., 
which is only allowed to stray 5^5 mm. in a thousand years, 
it is not much to boast of. The physical results do not 
spontaneously suggest any higher degree of freedom than 
this. But it seems to me that philosophical, psychological, 
and in fact commonsense arguments for greater freedom are 
so cogent that we are justified in trying to prise the door 
further open now that it is not actually barred. How can this 
be done without violence to physics ? 

If we could attribute the large-scale movements of our 
bodies to the "trigger action" of the unpredetermined 
behaviour of a few key atoms in our brain cells the problem 
would be simple; for individual atoms have wide indeter- 
minacy of behaviour. It is obvious that there is a great deal 
of trigger action in our bodily mechanism, as when the pent 
up energy of a muscle is released by a minute physical change 
in a nerve; but it would be rash to suppose that the physical 
controlling cause is contained in the configuration of a few 
dozen atoms. I should conjecture that the smallest unit of 
structure in which the physical effects of volition have their 
origin contains many billions of atoms. If such a unit behaved 
like an inorganic system of similar mass the indeterminacy 
would be insufficient to allow appreciable freedom. My own 
tentative view is that this "conscious unit" does in fact differ 
from an inorganic system in having a much higher indeter- 
minacy of behaviour simply because of the unitary nature 
of that which in reality it represents, namely the Ego. 

We have to remember (hat the physical world of atoms, 
electrons, quanta, etc., is the abstract symbolic representation 
of something. Generally we do not know anything of the 
background of the symbols we do not know the inner 
nature of what is being symbolised. But at a point of contact 
of the physical world with consciousness, we have ac- 
quaintance with the conscious unity the self or mind 
whose physical aspect and symbol is the brain cell. Our 


method of physical analysis leads us to dissect this cell into 
atoms similar to the atoms in any non-conscious region of 
the world. But whereas in other regions each atom (so far 
as its behaviour is indeterminate) is governed independently 
by chance, in the conscious cell the behaviour symbolises a 
single volition of the spirit and not a conflict of billions of 
independent impulses. It seems to me that we must attribute 
some kind of unitary behaviour to the physical terminal of 
consciousness, otherwise the physical symbolism is not an 
appropriate representation of the mental unit which is being 

We conclude then that the activities of consciousness do 
not violate the laws of physics, since in the present indeter- 
ministic scheme there is freedom to operate within them. 
But at first sight they seem to involve something which we 
previously described (p. 64) as worse than a violation of the 
laws of physics, namely an exceedingly improbable coin- 
cidence. That had reference to coincidences ascribed to 
chance. Here we do not suppose that the conspiracy of the 
atoms in a brain cell to bring about a certain physical result 
instead of all fighting against one another is due to a chance 
coincidence. The unanimity is rather the condition that the 
atoms form a legitimate representation of that which is 
itself a unit in the mental reality behind the world of 

The two aspects of human freedom on which I would lay 
most stress are responsibility and self-understanding. The nature 
of responsibility brings us to a well-known dilemma which 
I am no more able to solve than hundreds who have tried 
before me. How can we be responsible for our own good 
or evil nature? We feel that we can to some extent change 
our nature; we can reform or deteriorate. But is not the 
reforming or deteriorating impulse also in our nature? Or, 
if it is not in us, how can we be responsible for it? I will not 
add to the many discussions of this difficulty, for I have no 


solution to suggest. I will only say that I cannot accept as 
satisfactory the solution sometimes offered, that responsibility 
is a self-contradictory illusion. The solution does not seem 
to me to fit the data. Just as a theory of matter has to corre- 
spond to our perceptions of matter so a theory of the human 
spirit has to correspond to our inner perception of our spiritual 
nature. And to me it seems that responsibility is one of the 
fundamental facts of our nature. If I can be deluded over 
such a matter of immediate knowledge the very nature of 
the being that I myself am it is hard to see where any 
trustworthy beginning of knowledge is to be found. 

I pass on to another aspect of the freedom allowed under 
physical indeterminacy, which seems to be quite distinct from 
the question of Free Will. Suppose that I have hit on a piece 
of mathematical research which promises interesting results. 
The assurance that I most desire is that the result which I 
write down at the end shall be the work of a mind which 
respects truth and logic, not the work of a hand which 
respects Maxwell's equations and the conservation of energy. 
In this case I am by no means anxious to stress the fact (if it 
is a fact) that the operations of my mind are unpredictable. 
Indeed I often prefer to use a multiplying machine whose 
results are less unpredictable than those of my own mental 
arithmetic. But the truth of the result 7 x 11=77 lies in i ts 
character as a possible mental operation and not in the fact 
that it is turned out automatically by a special combination 
of cog-wheels. I attach importance to the physical un- 
predictability of the motion of my pen, because it leaves it 
free to respond to the thought evolved in my brain which 
may or may not have been predetermined by the mental 
characteristics of my nature. If the mathematical argument 
in my mind is compelled to reach the conclusion which a 
deterministic system of physical law has preordained that my 
hands shall write down, then reasoning must be explained 
away as a process quite other than that which I feel it to be. 


But my whole respect for reasoning is based on the hypothesis 
that it 15 what I feel it to be. 

I do not think we can take liberties with that immediate 
self-knowledge of consciousness by which we are aware of 
ourselves as responsible, truth-seeking, reasoning, striving. 
The external world is not what it seems; we can transform 
our conception of it as we will provided that the system of 
signals passing from it to the mind is conserved. But as we 
draw nearer to the source of all knowledge the stream should 
run clearer. At least that is the hypothesis that the scientist 
is compelled to make, else where shall he start to look for 
truth? The Problem of Experience becomes unintelligible 
unless it is considered as the quest of a responsible, truth- 
seeking, reasoning spirit. These characteristics of the spirit 
therefore become the first datum of the problem. 

The conceptions of physics are becoming difficult to under- 
stand. First relativity theory, then quantum theory and wave 
mechanics, have transformed the universe, making it seem 
fantastic to our minds. And perhaps the end is not yet. But 
there is another side to the transformation. Naive realism, 
materialism, and mechanistic conceptions of phenomena 
were simple to understand; but I think that it was only by 
closing our eyes to the essential nature of conscious experience 
that they could be made to seem credible. These revolutions 
of scientific thought are clearing up the deeper contradictions 
between life and theoretical knowledge. The latest phase 
with its release from determinism is one of the greatest steps 
in the reconciliation. I would even say that in the present 
indeterministic theory of the physical universe we have 
reached something which a reasonable man might almosi 



That's Shell that was!! 

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WE have seen that all knowledge of physical objects is 
inferential. The external world of physics is a universe 
populated with inferences. Familiar objects which we handle 
are just as much inferential as a remote star inferred from an 
image on a photographic plate or an "undiscovered" planet 
inferred from irregularities in the motion of Uranus. 
In the universe of inferences past, present and future appear 

, -7* <-.-. ^ --..--.-. . . l ., J,, 4, A 

indiscriminately, and it requires scientific analysis to sort 
them out. By a certain rule of inference, viz. the law of 
gravitation, we infer the present or past existence pfjjn 
invisible companion to a star; by an application of the same 
rule we infer the existence on Aug. 11, 1999, of a configura- 
tion of the sun, earth and mopjj, which corresponds to jijtotal 
e'dipse of the sun. In principle we have no reason to place 
greater confidence in the present inference than in the future 
inference; indeed it would generally be considered that we 
are less likely to have made a mistake about the eclipse. Both 
are of the same nature as the familiar inference that there is 
or was! ! a motor car "over yonder", which depends on 
our experience that light generally travels in straight lines, 
a law which is by no means so regularly obeyed as the law 
of gravitation. Thj^sliadow of the nioon on Cornwall in 
1999 is already in the world of inference. It will not change 
its status -wEen the vear 1000 arrives and we 


we shall merely substitute one method of inferring 
the shadow for another. The shadow will always be an 
inference. By the shadow I here mean the entity or condition 
in the physical world, viz. a comparatively quiescent state 
of the aether, not the sensory perception of darkness in a 
number of human and animal minds. 

Of particular importance for the problem of determinism 
are our inferences about the past. Strictly speaking our most 
direct inferences from sight, sound and touch all relate to a 
time slightly antecedent to the sensation. To obtain an 
inference as to the present state of things we have to combine 
them with our general inferential knowledge of the con- 
tinuity of phenomena obtained from other experiences. But 
there are cases in which the time lag is more considerable, 
or for other reasons the argument of continuity does not 
apply. Suppose that we wish to determine the chemical 
constitution of a certain salt. We put it in a test tube and 
apply various reagents, and from the phenomena observed 
reach the conclusion that it was silver nitrate. It is no longer 
silver nitrate after our treatment of it. The property which 
we infer is not that of "being X" but of "having been X". 
We say in fact "That's X that was!!" I will call this 
retrospective inference. 

We noted at the outset (p. 76) that in considering deter- 
minism the alleged causes must be challenged to produce 
their birth-certificates, so that we may know whether they 
really were pre-existing. Retrospective inference is par- 
ticularly dangerous in this connection because it involves 
antedating a certificate. The experiment above mentioned 
certifies the chemical constitution of a substance, but the date 
we write on the certificate is earlier than that at which we 
became assured of the composition. 

To show how retrospective inference might be abused, 
suppose that there were no way of learning the chemical 
composition of a substance without destroying it. By 


hypothesis a chemist would never know until after his 
experiment the composition of the substance he had been 
handling, so that the result of every experiment must be 
unforeseen. Must he then admit that the science of chemistry 
is chaotic ? A man of resource would override so trifling an 
obstacle. If he were discreet enough never to say beforehand 
what his experiment was going to demonstrate, he might 
give edifying lectures on the uniformity of Nature. He puts 
a lighted match in a cylinder of gas and the gas burns 
"There you see that hydrogen is inflammable". Or the 
match goes out "That proves that nitrogen does not sup- 
port combustion". Or the match burns more brightly 
"Oxygen feeds combustion". "How do you know it was 
oxygen ? " "By retrospective inference from the observation 
that the match burns more brightly." And so the experi- 
menter passes from cylinder to cylinder, and the match does 
now one thing and now another, thereby beautifully demon- 
strating the uniformity of Nature and the determinism of 
chemical law! 

If by retrospective inference we infer causal characters at 
an earlier date and then say that those characters invariably 
produce at a future date the manifestations from which we 
made the inference, we are working in a vicious circle. The 
connection is not causation but definition, and we are not 
prophets but tautologists. We must not mix up the genuine 
achievements of scientific prediction with this kind of 
charlatanry, nor the observed uniformities of Nature with 
those so easily invented by our imaginary lecturer. If we are 
to avoid vicious circles we must refuse to recognise purely 
retrospective characteristics those which are never found 
as existing but always as having existed. If they do not 
manifest themselves until the moment that they cease to 
exist they can never be used for prediction except by those 
who prophesy after the event. 

Chemical constitution is not one of these retrospective 


characters, although it is often inferred retrospectively. The 
fact that silver nitrate can be bought and sold shows that 
there is a property of being silver nitrate as well as of having 
been silver nitrate. If a property can be assigned retro- 
spectively the method of sampling usually enables us to 
assign the same property simultaneously. We divide a giveq 
substance into two parts, analyse one part (destroying it if 
necessary) and show that its constitution has been X\ then it 
is usually a fair inference that the constitution of the other 
part 15 X. If that method were universally applicable there 
would be no danger of introducing into physics characters 
which have only a retrospective existence. But the method 
of sampling is inapplicable when we consider those charac- 
teristics which are supposed to distinguish one atom from 
another; for the individual atom cannot be divided into two 
samples, one to analyse and one to preserve. So it is in the 
domain of atomic physics that the confusion caused by 
retrospective inference has arisen. 

It is known that potassium consists of two kinds of atoms, 
one kind being radio-active and the other inert. Let us call 
the two kinds K a and K^ . If we observe that a particular atom 
bursts in the radio-active manner we shall infer that it was 
a K a atom. Can we say that the explosion was predetermined 
by the fact that it was a K a and not a K0 atom? On the 
information stated there would be no justification at all; 
K a is merely an antedated label which we attach to the atom 
when we see that it has burst. We can always do that however 
undetermined the event may be which occasions the label. 
When I see at Cambridge station an assemblage of parcels 
from different parts of the country all bearing the Cambridge 
label, I infer an efficient organisation. But that is on the 
supposition that the parcels were labelled when they were 
dispatched. It is no proof of organisation if someone has 
gone round sticking Cambridge labels on everything that 
happened to turn up at Cambridge. Actually, however, we 


have information which shows that the burst of the potassium 
atom is not undetermined. Potassium is found to consist oJ 
two isotopes of atomic weights 39 and 41 ; and it is believed 
that 41 is the radio-active kind, 39 being inert. It is possible 
to separate the two isotopes and to pick out atoms known to 
be K 4 i . Thus K 4 i is a contemporaneous character and can 
legitimately predetermine the subsequent radio-active out- 
burst; it replaces the character K a which was found retro- 

So much for the fact of outburst; now consider the time 
of outburst. Nothing is known as to the time when a 
particular K 4I atom will burst except that it will probably 
be within the next billion years. If, however, we observe 
that it bursts at a time t we can ascribe to the atom the 
retrospective character K, , meaning that it had (all along) 
the property that it was going to burst at time t. Now 
according to modern physics the character K* is not mani- 
fested in any way is not even represented in our mathe- 
matical description of the atom until the time t when the 
burst occurs and the character K* having finished its job 
disappears. In these circumstances K t is not a predetermining 
cause. Our retrospective label adds nothing to the plain 
observational fact that the burst occurred without warning 
at the moment t; it is merely a device for ringing a change 
on the tenses. 

The super-intelligence imagined by Laplace was able to 
foresee the whole future; but the proviso was that he must 
be acquainted with all the conditions prevailing at a given 
instant. How much does this proviso include ? If it includes 
all retrospective characters that might be attributed, that is 
to say all that might be inferred by retrospective inference 
from what actually will happen in the future, he is using the 
future to predict the future. He can predict the exact time 
of break-up of the radio-active atom if he is told the character 
]* of the atom; but that is just the same as being told the 


time of break-up. Clearly then we must exclude retrospective 
characters. And if Laplace's being cannot predict the future 
without them, we turn instead to the being whom secondary 
law has substituted, whose vision of the future is incomplete 
and nowhere reaches entire certainty but, so far as it goes, 
has the merit of being genuine foreknowledge. 

We have seen that a retrospective character is a device for 
ringing the changes on the tenses. Such a device may be 
useful in systematising our knowledge. By replacing the 
undetermined events of the future by indeterminate characters 
ascribed to the present we telescope the whole course of 
events into one apparently instantaneous scheme. The in- 
determinism of the future is accordingly made to appear as 
an indeterminacy of the present. From the purely philo- 
sophical point of view this is a confusing way of expressing 
things; but that is of no particular concern to the scientist 
who is willing to adopt any device which helps him to get 
on with the job of formulating and applying the laws which 
decide the recurrencies of experience in an indeterministic 
world. In the next section we shall see how this device 


In 1927 W. Heisenberg formulated an important principle 
which defines clearly the amount of indeterminism in the 
accepted system of physical law. It is called the Principle of 
Uncertainty or sometimes the Principle of Indeterminacy. 
This was not the origin of the change over of physics from 
determinism to indeterminism; but it called attention to the 
existing indeterminism in a way which could scarcely be 
overlooked even by those least attentive to the philosophy 
of science. 

Laplace imagined an intelligence who " would include in 
one and the same formula" the movements of all the bodies 



in the universe. But to include them in a formula is not 
necessarily the same thing as to know them. An algebraic 
symbol may stand for a known or for an unknown quantity. 
So when we have a formula which professes to give exactly 
the future position of an object, the question arises whether 
it is given in terms of known or of unknown symbols. 
Heisenberg's principle tells us that just half of the symbols 
represent knowable quantities and the other half represent un- 
knowable quantities. The unknowable quantities correspond 
to retrospective characters. By inventing such characters we 
make the future appear determinate; but they do not actually 
predetermine the future because they are themselves in- 
determinate until the future events have taken place. 

This may seem a rather artificial way of describing the 
indeterminism of the future, but it shows that there is method 
even in indeterminacy. Looking first at the consequences of 
the indeterminacy we find that some phenomena are pre- 
dictable with practical certainty whereas others are almost 
wholly spontaneous, but we do not discover any simple rule. 
But looking at the causes of the indeterminism (if an Irishism 
may be allowed) we find that what is lacking to secure a 
complete and certain prediction of the whole future is always 
just half of the total data that would be needed. The data are 
paired in such a way that for each datum or character 
inferable from manifestations up to a given instant there is 
a symmetrical datum a retrospective character which is 
not inferable until later; and without both data the exact 
prediction is impossible. In this sense the future is half linked 
to the past and half detached from it. 

This is often expressed in the form that there is an inter- 
ference between our experimental attempts to determine the 
two data. That has the disadvantage that it raises the question 
of our skill and ingenuity. I do not want to reopen the whole 
question of determinism versus indeterminism discussed in 
the last chapter; my purpose now is not to defend but to 


examine the implications of the indeterminism contained in 
the existing physical theory. According to that theory there 
is an incompatibility of the two data inherent in their own 
nature and not due to the operations of an intervening 

Let us consider an isolated system. It is part of the universe 
of inference, and all that can be embodied in it must be 
capable of being inferred from the influences which it broad- 
casts. Whenever we state the properties of a body in terms 
of physical quantities we are imparting knowledge as to the 
response of various external indicators to its presence and 
nothing more. A knowledge of the response of all kinds of 
objects would determine completely its relation to its en- 
vironment, leaving only its un-get-at-able inner nature which 
is outside the scope of physics.* Thus if the system is really 
isolated, so that it has no interaction at all with its sur- 
roundings, it has no properties belonging to physics but only 
an inner nature which is beyond physics. So we must modify 
the conditions a little. Let it for a moment have some 
interaction with the world exterior to it; the interaction 
starts a train of influences which may reach the nerves and 
brain of an observer and become translated into sensory 
experience. From this one signal the observer can draw an 
inference about the system, i.e. he can fix the value of one 
of the symbols describing the system or fix an equation 
connecting several such symbols. To determine more symbols 
there must be further interactions resulting in sensory ex- 
perience, one for each symbol fixed. 

It might seem that in time we could fix all the symbols in 
this way, so that there would be no undetermined symbols 
in the description of the system and no unknown quantities 
in the equations which profess to foretell the future. But it 
must be remembered that the interaction which disturbs the 
external world by sending a signal through it also reacts on 
* The Nature of the Physical World, p. 257. 



the system. There is thus a double consequence; the inter- 
action starts a signal informing us that the value of a certain 
symbol q in the system is qi, and at the same time it alters to 
an unknown extent the value of another symbol p in the 
system. If we have learnt from former signals that the value 
of p is pi, our knowledge ceases to apply, and we must start 
again to find the new value of p. Presently there is another 
interaction which tells us that p is now p?, ; but the same 
interaction knocks out q, so that we no longer know its value. 
The observer is like the comedian with an armful of parcels; 
each time he picks up one he drops another. 

It is of the utmost importance for prediction that the 
quantity which is upset by the interaction is not the quantity 
we are inferring but a paired quantity. If the signal taught 
us that at the moment of the interaction q was qi but that as 
the result of the interaction it has been changed to an 
unknown extent, we should never have anything but retro- 
spective knowledge like my imaginary chemistry lecturer 
who always destroyed his substances in the act of ascertaining 
their composition. The pairing allows us to have contem- 
poraneous knowledge of half the symbols but never more 
than half. This, of course, is not an a priori rule of indeter- 
minacy. Heisenberg's discovery was that it is the rule of the 
indeterminacy which applies to the physical universe. 

Heisenberg's principle contains something more. We have 
been contemplating only two alternatives, viz. that the value 
of a symbol q is either known or unknown. But it may be 
partially known; that is to say, we may know it within 
certain limits of accuracy and with a certain degree of 
probability. If one of two paired symbols is known with 
certainty and accuracy the other must be altogether unknown ; 
but if one is partially known the other may be partially 
known. For such partial knowledge Heisenberg's principle 
gives the rule that the uncertainty (or standard deviation) 
of the quantity q multiplied by the uncertainty of the paired 


quantity p is of the order of magnitude of Planck's constant h. 
The product of the two standard deviations is of the order 
of magnitude of one quantum. 

The general Definition jof_ ihejsaired symbols is rather a 
technical matter ; they are called coordinates _and momenta, each 
coordinafeliavmg aTmomentum paired with it. To show the 
results of the principle we will consider the position and 
velocity of an electron. We can fix the position of an electron 
to within about -ooi mm. and (simultaneously) the velocity 
to within about i km. per sec.; or we can fix the position 
to -oooi mm. and the velocity to 10 km. per sec.; or the 
position to -ooooi mm. and the velocity to 100 km. per sec.; 
and so on. We can divide the uncertainty as we like, but it 
cannot be got rid of. The secret is that if by our experimental 
arrangements we persuade the electron to send us a very 
sharp signal of its position, its velocity (which it had pre- 
viously signalled) is altogether upset by the reaction. But it 
is possible to compromise. If we allow the electron to send 
a less precise indication of its position, the reaction is less 
intense and the velocity does not get so bad a knock. 

This combination of uncertainty is actually embodied in 
the present theoretical picture of an electron. Nowadays we 
represent an electron not by a corpuscle but by a packet of 
waves ; and the notion of exact position coupled with exact 
velocity which applies to a corpuscle does not apply to a 
packet of waves. So if we describe something as having 
exact position and exact velocity, we cannot be describing 
an electron; just as (according to Bertrand Russell) if we 
describe a person who knows what he is talking about and 
whether what he is saying is true, we cannot be describing 
a pure mathematician. It is therefore not a question of lack 
of skill on our part, or a casual difficulty in the experimental 
handling of these minute objects, or a perverse delight of 
Nature in tantalising us. If ever the day arrives when by 
improved technique an experimenter measures the position 


and velocity of an electron with greater accuracy than 
Heisenberg's principle admits, the present quantum theory 
will join the limbo of forgotten theories. 

We might spend a long time admiring the detailed 
working of these paired uncertainties which prevent us from 
knowing more than we ought to know. But I do not think 
you should look upon it as Nature's device to prevent us 
from seeing too far into the future. The future is not pre- 
determined, and Nature has no need to protect herself from 
giving away plans which she has not yet made. But the 
mathematician has to protect his equations from making 
impossible predictions. It commonly happens that when we 
ask silly questions, mathematical theory does not directly 
refuse to answer but gives us an oracular answer like o/o out 
of which we cannot wring any meaning. Similarly when we 
ask where the electron will be to-morrow, the mathematical 
theory does not give the straightforward answer "It is im- 
possible to say because it is not yet decided", because that is 
beyond the resources of an algebraic vocabulary. It gives us 
an ordinary formula in x's and /s, but it makes sure that we 
cannot possibly find out what the formula means until 


The Principle of Uncertainty has the same kind of position 
in physics as the Principle of Relativity. Both have arisen 
from the discovery of what appeared at first to be a tantalising 
limitation of our resources of observation. The theory of 
relativity originated in the discovery that we cannot observe 
the motion of ourselves or of anything else relative to the 
aether. That seemed at first to be a casual obstacle in our 
search for truth; but it is now realised that our failure was 
due to the fact that we were looking for something which did 
not exist. Since then we have been on the look out for other 
pitfalls of the same kind. We must make sure that the quan- 


dries or characters that we speak about are directly or 
indirectly definable in terms of experience otherwise our 
words convey no meaning. It was suspected that something 
of this kind was at the root of the difficulties of the old 
quantum theory; but the precise point of failure of our 
definitions eluded detection. The following passage, written 
shortly after the great awakening brought about by the 
theory of relativity, will illustrate the thought of the time.* 

I should be puzzled to say off-hand what is the series of opera- 
tions and calculations involved in measuring a length of lO" 1 ^ cm. ; 
nevertheless I shall refer to such a length when necessary as though 

it were a quantity of which the definition is obvious 1 may 

be laying myself open to the charge that I am doing the very 
thing I criticise in the older physics using terms that have no 
definite observational meaning, and mingling with my physical 
quantities tilings which are not the results of any conceivable 
experimental operation. I would reply By all means explore 
this criticism if you regard it as a promising field of inquiry. I here 
assume that you will probably find me a justification for my 
lO" 1 ^ cm. ; but you may find that there is an insurmountable am- 
biguity in defining it. In the latter event you may be on the track 
of something which will give a new insight into the fundamental 
nature of the world. Indeed it has been suspected that the per- 
plexities of quantum phenomena may arise from the tacit assump- 
tion that the notions of length and duration, acquired primarily 
from experiences in which the average effects of large numbers 
of quanta are involved, are applicable in the study of individual 
quanta. There may need to be much more excavation before we 
have brought to light all that is of value in this critical considera- 
tion of experimental knowledge. Meanwhile I want to set before 
you the treasure that has already been unearthed in this field. 

The excavation has proceeded and has not revealed any- 
thing wrong with lo^s cm., nor with very minute measure- 
ments of other quantities. The pitfall was just a stage more 
subtle than that which relativity theory had exposed. It is 

* Eddington, Mathematical Theory of Relativity p. 7 


the combination of two exact measurements which has proved 
to have no definable meaning in terms of experience, although 
either measurement alone would express something definite. 

The remedy adopted by relativity theory was simple; it 
expelled the quantities such as " velocity through aether" 
which were found to have no meaning. Thus purged, the 
physical universe became identified with the knowable. But 
the same treatment could not be applied to two quantities 
which play "Box and Cox". We have had to give up the 
attempt to define an objective world which corresponds 
exactly to what is potentially knowable. We have instead a 
universe which is just half knowable, and we are free to 
choose which half we shall set about knowing. That at least 
is how it appears when described in terms of our ordinary 
epistemological outlook. Equivalently, by the substitution 
of two quantities partially known for one quantity known 
and the other unknown, we reach the outlook of wave 
mechanics. What is knowable, i.e. inferable from experience, 
is a distribution of probability; we infer, not a series of 
events in the objective universe but the degree of probability 
of all possible events in the objective universe. Thus between 
the universe inferable from experience and the objective 
universe there is interposed the rather baffling conception of 
probability, which we shall try to understand in the next 

I have said that the indeterminism of the future applies to 
all phenomena, although for some it may be practically 
insignificant. Perhaps you will think this statement too 
sweeping. Referring again to the isotopes of potassium, it 
is not predetermined whether a million years hence a given 
atom of the radio-active isotope K 4I will or will not have 
broken up. On the other hand K 39 is non-radio-active and 
has not enough energy to explode. Then (it will be said) 
there is at least one predetermined fact about its future; we 
can predict without any indeterminism that a million years 


hence it will not have broken up. I am not going to object 
that you are pressing my statement unfairly. I meant just 
what I said though I must ask you not to look on that as 
a precedent.* Strictly speaking there is no such thing as a 
K 39 atom, but only an atom which has a high probability of 
being K 39 . Such an atom should contain 39 protons within 
a comparatively small nucleus; but a proton in modern 
physics (like an electron) is never anywhere quite definitely 
though it may have a higher probability of being in one 
place than another. Thus we can never get beyond a high 
probability of 39 protons being collected together. It is 
impossible to trap modern physics into predicting anything 
with perfect determinism because it deals with probabilities 
from the outset. 


Heisenberg's principle has a very curious consequence when 
it is applied to an angular position and to a corresponding 
angular momentum. These are paired quantities, and the 
product of their uncertainties is a quantum, i.e. Planck's 
constant h. Consequently if we want to know the angular 
momentum of a system very accurately there must be a very 
wide uncertainty in our knowledge of the angular position 
or orientation of the system. But a difficulty arises. How- 
ever careless we are, we cannot make a mistake of more than 
360 in laying down an orientation, or in the usual circular 
measure the greatest possible uncertainty in angle is 2-rr. 
Therefore by Heisenberg's rule the least possible uncertainty 
in angular momentum is h/2ir. This forms a kind of discrete 
unit of angular momentum. We cannot distinguish differences 
of spin finer than this unit. When we describe changes of 
spin of an atom such changes must amount to one or more 
units ; for a smaller change has no meaning definable in terms 
of experience. We have thus to picture a kind of change 

* See p. 279. 


which can only occur by jumps of a whole unit. This is one 
way of realising the origin of the orbit "jumps" of an 
electron, which were such a mysterious feature of the older 
quantum theory. I daresay that viewed in this way they 
become even more mysterious; but at least they are now 
seen to be a special case of a very general principle which 
covers the whole indeterminacy of physics, and not merely 
a sporadic phenomenon inside an atom. 

It also follows that in the small-scale systems we cannot 
separate geometry from dynamics. As soon as we introduce 
into our picture of the world anything possessing orientation, 
it automatically begins to spin one way or the other. To say 
that there is definitely no spin would be to claim an accuracy 
of knowledge which we have seen to be impossible. The 
most "restful" system we can contemplate is one equally 
likely to have any value of the spin up to half a unit in either 
direction. Knowing then the probability distribution, we 
can compute the average energy of spin of a large number 
of these restful atoms; in macroscopic physics the average is 
all that concerns us. The more complicated the system, the 
greater will be the number of directions or orientations 
defined in our picture of it; and since each of these has its 
own uncertainty of spin, there is on the average a considerable 
amount of angular momentum present which is of this 
irreducible kind. In short, in ascribing geometry to the 
system we are compelled by the Uncertainty Principle to 
ascribe to it energy of constitution. Observation may show 
us that more than this minimum energy is present. Such 
additional energy is called energy of excitation-, it may be 
radiated or otherwise passed from system to system. 

This kind of application of the Uncertainty Principle has 

been used by F. A. Lindemann* to explain a number of 

striking results and paradoxes of the quantum theory. I 

do not think we need trouble much about the rigour or 

* The Physical Significance of the Quantum Theory. 


precision of the method. The Uncertainty Principle arises out 
of the wave constitution of electrons and protons; and those 
who put "safety first" will naturally proceed to these results 
by rigorous solution of the wave equations. In this book we 
are not prepared to follow the detailed progress of the 
mathematician, who is cautiously finding his way through 
the maze of passages in the edifice of wave mechanics; so we 
are grateful for a window through which we can catch a 
glimpse of one or two of the interesting rooms. 

The guiding principle can perhaps be expressed as follows. 
It would be illogical to admit as a constituent of the external 
world a carbon atom whose properties were inconsistent 
with its being known to be a carbon atom illogical because 
the name refers not to its inner nature (which is outside 
physics) but to its manifestations. Therefore when we speak 
of a carbon atom, we imply that it has undergone the reactions 
which are involved in signalling through its surroundings 
that it is such a system as a carbon atom is defined to be, 
namely a nucleus with six satellite electrons. Thus the men- 
tion of a carbon atom implies inter alia that it has been possible 
to count the number of electrons and make reasonably sure 
that the number is six not five. Let us imagine ourselves 
counting them: "One, two, three, four, five, Now is that 
a sixth, or have I already counted it?" You cannot count 
unless the objects have some degree of fixity of position, as 
those who induce slumber by counting sheep in a green field 
are well aware. But the more closely you fix the positions 
the bigger the uncertainty of momentum. So when you 
consider six electrons in an atom you have to attribute (on 
the average) more angular momentum than is involved in 
connecting each individually to the atom. This is a con- 
sequence of introducing enough distinction of position for 
it to be externally manifest that six electrons are at work and 
not fewer. In this kind of way Lindemann arrives at the 
Exclusion Principle by which each electron in the atom must 


have a separate quantum orbit sufficiently differentiated from 
the orbits of the others. 

My own interest in this method is bound up with its 
application to "finite but unbounded space". Here again the 
rigorous demonstration rests on the equations of wave 
mechanics; but the Uncertainty Principle is useful for a 
preliminary insight. We have seen that in the theory of 
relativity space-time has a natural curvature, so that three- 
dimensional space curves round and closes up analogously 
to the two-dimensional surface of a sphere. It is evident that 
in a finite space of this kind we cannot make so big a mistake 
about the position of anything, as we could in infinite open 
space. It is impossible to be more than 12,000 miles out in 
locating an inhabitant of the earth; and similarly there is an 
upper limit (some thousands of millions of light years) to the 
possible error in locating an electron or any other inhabitant 
of our more or less spherical universe. Just as before, this 
upper limit to the uncertainty of position implies a lower 
limit to the uncertainty of momentum; so that in the most 
favourable case, when we know nothing at all about the 
position of the electron except that it is somewhere (i.e. within 
the limits of the universe), it must have a small uncertainty 
of momentum. A large number of electrons and protons 
will accordingly possess a certain irreducible average 
momentum and energy. 

This result of wave mechanics throws further light on the 
meeting point of relativity theory and quantum theory 
(p. 48). In relativity theory mass (or energy) and momentum 
are associated with curvature of space-time, and indeed are 
identified with the measures of certain components of the 
curvature; the law of conservation of energy and momentum 
and the gravitational effect which one mass exerts on another 
are deducible from this identification. On the other hand 
quantum theory has treated the energy and momentum of 
a particle empirically without revealing that they have any 


connection with curvature. We now see that energy and 
momentum will arise out of curvature of space according to 
the principles of quantum theory as well as according to the 
principles of relativity theory. The modus operandi is that 
curvature limits the extent of space available for the particle 
to roam over and so limits our ignorance of its position; 
hence by Heisenberg's principle there is introduced a 
minimum uncertainty and therefore a non-zero average 
value of the energy and momentum. 

The further development of this theory of the origin of 
mass must be postponed to Chapter XL 


It is remarkable that a science which began with the consideration of 
games of chance, should have become the most important object of 
human knowledge. LAPLACE, Theorie Analytique des Probability 


ABOUT the beginning of the nineteenth century the mathe- 
matical theory of probability attained great prominence 
through the writings of Laplace, Gauss and other famous 
mathematicians. It has had many applications in physical 
science. At first it was almost wholly confined to the treat- 
ment of errors of observation especially in astronomy, 
which seems to have enjoyed the doubtful distinction of 
being the subject which provides most scope for a theory of 
errors. With the rise of thermodynamics and the analysis of 
matter into great numbers of independent particles moving 
at random, probability has entered more intimately into the 
fundamental problems of physics. To-day the pre-eminent 
symbol in wave mechanics, the mysterious $ which the 
quantum physicist pursues from equation to equation, is in 
so far as we may define the indefinable identified with 
probability. In the most modern theories of physics prob- 
ability seems to have replaced aether as "the nominative of 
the verb 'to undulate'". 

Since it is so often necessary to refer to probability in these 
lectures, I have thought it well to devote a chapter to this 
much debated subject. We ought at least to clarify our ideas 
sufficiently to use the conception of probability consistently 
and logically in its scientific application. 

When a word in everyday use is adopted as an exact 


scientific term it does not always retain its everyday meaning. 
For example, in mechanics work is a technical term having 
a meaning by no means coextensive with our ordinary notion 
of work. Scientifically no work is done unless something is 
moved. The acrobat who stands at the base of a tableau, with 
the other members of the troupe supported gracefully on his 
shoulders, does no work. Similarly it must not be expected 
that probability when used as an exact term in mathematics 
and physics will retain all the shades of meaning that it may 
have in ordinary conversation. As a technical scientific term 
it denotes something to which a definite numerical measure 
can be attributed; to secure this definiteness we must sacrifice 
some of the looser implications of probability. 

Before proceeding to the scientific and mathematical de- 
finition let us examine the most common use of the word. 
We speak of the probability that a prisoner is guilty, or the 
probability that a certain course of action will be successful. 
The probability is rated as "high" or "low", but there is not 
usually any ground for assigning a numerical measure to it. 

In this case probability refers to the strength of our ex- 
pectation or belief. The probability of an event refers to the 
strength of our expectation that it will occur; the probability 
of a theory refers to the degree of confidence that a right- 
thinking person would have in it. I do not think there is any 
difference of substance between the two statements : (a) on 
the evidence it is highly probable that the prisoner is guilty, 
and (b) a right-thinking person would form from the evidence 
a strong belief that the prisoner is guilty. It must always be 
recognised that, both in the ordinary and in the scientific use 
of probability, the probability is dependent on or "is relative 
to" the information supplied; for additional information is 
likely to modify our expectation of an event or our confidence 
in a belief. In no circumstances is probability an absolute 
attribute of an event or a belief. 

The question arises whether we can use the strength of 


belief as a measure, or as the basis of a measure, of the 
probability. In my view this is impossible. At any rate the 
measurement of probability employed in mathematics and 
physics has an altogether different basis, as we shall see. 

One difficulty in employing strength of belief as a measure 
of probability is that an expectation or belief has partly a 
subjective basis. We have agreed that it depends (and ought 
to depend) on the information or evidence supplied; but in 
addition the strength of the expectation depends on the 
personality of the man who weighs the evidence. We try to 
remove this subjective clement by saying that the true prob- 
ability corresponds to the judgment of a "right-thinking 
person"; but how shall we define this ideal referee? We do 
not mean a perfectly logical person, for there is no question 
of making a strictly logical deduction from the evidence; if 
that were possible the conclusion would be a matter of 
certainty not probability. We do not mean a person gifted 
with second-sight, for we want to know the probability 
relative to the information stated and not relative to occult 
information. We do not particularly mean k person of long 
experience in similar judgments, for he is likely to use his 
past experience to supplement surreptitiously the information 
on which the judgment of probability is ostensibly based. 
Apart from the obvious definition of a right-thinking person 
as "someone who thinks as I do" (which is probably the 
definition at the back of our minds) there seems to be no way 
of defining his qualities. 

There are, of course, occasions when all sensible persons 
agree in rating the probability of one event as high and of 
another event as much lower; so that, if we do not attempt 
too precise a classification, the question of subjectivity of 
judgment does not arise. But there is no reason to think that 
these probabilities can be graded systematically in order of 
magnitude. It has been maintained by some writers that 
probability always has a numerical measure even when the 


word is used in this elementary way; and that the beliefs of 
a right-thinking person could ideally be arranged in a unique 
sequence in order of intensity. I rate this on a level with the 
view that to a person with a right sense of humour all jokes 
can be arranged in a unique sequence in order of funniness. 
We conclude then that the most elementary use of the 
word probability refers to strength of expectation or belief 
which is not associated with any numerical measure. There 
can be no exact science of these non-numerical probabilities 
which reflect personal judgment. They form an important 
element in our outlook as do many other things which do 
not come within the scope of exact measurement. We act on 
such probabilities, and we are justified in so acting. Man is 
not just a logic factory. He is an adventurer, and the taking 
of risks is a condition of life. Expectations are sometimes 
fulfilled and sometimes disappointed. But Man goes on 


We turn now to probabilities which admit of numerical 
measurement. Numerical estimates of probability are often 
made in ordinary conversation; e.g. "It is 5 to i that the 
prisoner is guilty". Here die intention is to give a general 
impression of the strength of one's belief, but no coherent 
explanation can be given as to why the measure number 5 
was selected. Sometimes an expression of this form does not 
really refer to anything that could properly be called prob- 
ability but concerns a proposed financial transaction. But 
other examples can be given in which the numerical prob- 
ability has been calculated in a systematic way, and we are 
guided by these in formulating the scientific definition of 
probability. As an example of a probability whose measure 
is definite and commonly recognised we take the statement 
"The probability is | that my next throw with the dice will 
be an ace". 



Let us first find out precisely what this statement means. 
Like many common statements the meaning is not to be 
discovered by examining the grammatical structure of the 
sentence. The best way of realising the meaning is to consider 
what evidence we should accept as proving or supporting 
the statement. Ostensibly it is a statement about "my next 
throw"; it would therefore seem natural to test its truth by 
making my next throw. But it is well known that that would 
provide no evidence one way or the other. The ace may turn 
up, or it may not; in either case there is no reason to change 
our opinion as to whether the odds against it were correctly 

A recognised test would be to throw the dice 6000 times. 
If the number of aces thrown is reasonably near 1000, that is 
regarded as satisfactory confirmation that the probability is ~. 
If it is, say, 1230, that is an almost conclusive disproof. 

Thus, although the statement refers to my next throw, its 
meaning is not specially connected with my next throw. 

Verbally the statement refers to a particular event; but its 
meaning refers to a class of events of which the particular 
event is one member. Thus numerical probability is a com- 
munal property, acquired through membership of a class. 
The statement 

The probability is p that an event a has an outcome e 
has to be translated 

The event a is a member of a certain class of events A, and the 
proportion of events in the class A which have an outcome e is p. 

The proportion of events in a given class which have an 
outcome e is generally called the frequency of e in that class. 
Thus a numerical measure frequency belonging to a class is 
verbally transferred to an individual member of that class 
and renamed probability. 

9y this definition we introduce a probability which is not 


based on strength of belief; it denotes simply the proportion 
of events with a given outcome in a defined class. We do not 
say that there is no connection between this kind of prob- 
ability and strength of belief; for the frequency of success 
will, like any other relevant information, be taken into 
account in forming a belief in the rather indefinite way in 
which beliefs arc formed. Certain beliefs may be mainly, or 
even wholly, based on a numerical probability. But there is 
no mathematical connection between the probability and the 
belief, for the passage from evidence to belief is not along 
mathematical lines. 

We have examined two common uses of the word prob- 
ability, the one a non-numerical probability associated with 
strength of belief or expectation, the other a numerical 
probability associated with frequency in a class. Both are 
well established in the language, and we can scarcely forgo 
either of diem. Both may be introduced in a single sentence. 
If the dice have not been tested we are not sure that they are 
true, and therefore we are not sure that the frequency of an 
ace turning up is - 6 . According to circumstances we may rate 
it as rather probable, highly probable, nearly certain, etc., 
that the dice are true. Then our statement will be "It is rather 
probable that the probability of my throwing an ace at the 
next throw is -". Here the first is a non-numerical prob- 
ability referring to the strength of belief; the second is a 
numerical probability or frequency. The second is the 
probability that has been taken over into science where it is 
used as a technical term; but the scientist cannot monopolise 
the language, and he must at times also use the word with 
the other non-technical meaning. 

Failure to distinguish the two usages has often caused 
obscurity in treating the subject. The common idea is that, 
since probability signifies uncertainty, a statement like the 
foregoing which contains two uncertainties ought to be 
reducible to simpler terms. But numerical probability is not 



an uncertainty; it is an ordinary physical datum the fre- 
quency of a certain characteristic in a class. Our knowledge 
of it may be uncertain, but so too is our knowledge of many 
other physical data. The statement "It is rather probable that 
the probability is. . . " is no more objectionable than the 
statement "It is rather probable that the solar parallax 

Normally the class of events A consists of, or at least 
includes, events which have not yet occurred, and the 
frequency of the outcome e is deduced from theory and not 
from actual statistics. This theoretical information about the 
class is not furnished by the theory of probability. For 
example, certain operations such as shuffling are supposed to 
give certain results with equal frequency. Again, it is often 
assumed that certain events will in the future occur with the 
same frequency as they have been observed to do in the past. 
The study of probability is often distracted by a discussion 
as to whether we have any proof of these assumptions. But 
the function of probability theory is to utilise such infor- 
mation, not to supply it. When once it is realised that there 
is nothing illogical in a numerical probability being itself 
only probable, we can utilise any reasonable belief as to the 
frequency of events and so determine "a reasonably probable 
probability"; just as we may use a reasonable belief as to 
the cause of the recession of die spiral nebulae and so deter- 
mine a reasonably probable cosmical constant. 

You will see that I do not discuss why, after having 
ascertained that an event belongs to a class containing 9 
successes to I failure, we generally form a fairly confident 
expectation that it will occur. I do not think this can be 
discussed apart from the formation of expectations based on 
other types of information. This is no doubt an important 
aspect ot the subject of probability , but it is scarcely within 
our province. If we maintained, as some have done, that 
scientific (numerical) probability is the basis of all rational 


belief other than strict logical deduction,* thereby annexing 
the whole subject of inference to the mathematical theory of 
probability, it would be necessary to go into the matter 
further. But that is not the position here adopted. 


We have seen that the probability assigned to an event is a 
property of a class of events. Usually the class is not directly 
mentioned in our statement; but there must be an implicit 
understanding, since otherwise the probability would be 
indeterminate. Thus I would say that the probability that 
Mussolini was born on a Friday is ^; the understanding is that 
his birth is assigned to the class of all human births, and 
I believe (though I may be mistaken) that human births are 
equally frequent on all days of the week. You may have 
looked up the date and found it to be, say, Tuesday; if so, 
you will assign it to the more limited class of human births 
which have occurred on a Tuesday, and say that the prob- 
ability is o. We are both right. The probability relative to 
the information in my possession is ^; relative to the greater 
information in your possession, it is o. 

This shows how probability comes to be relative to the 
information supplied. The information is used to define the 
class to which the event in question is assigned; additional 
information causes us to re-define the class. In this way more 
than one probability may belong to the same event. What 
is the probability that it will rain to-morrow (April 19)? 
This may refer to the frequency of rain on April 19 in all 
years; or to the frequency with which meteorological con- 
ditions similar to those now prevailing are followed by rain 
on the next day; or to a class satisfying both conditions. 

* Logical deductions can be regarded as a special case corresponding 
to probability i, i.e. certainty. 


There are three or more numerical probabilities attached to 
the same event a quite permissible situation. 

The question arises which of these is the practical prob- 
ability the one by which we should be guided when we 
stand hesitating by the umbrella stand. If the probabilities 
were certain probabilities and not merely probable probabilities, 
there is no doubt that the third should be chosen the one 
which embraces all the available information. This may be 
seen in the following way. Suppose that the frequency of 
rain on April 19 is quite definitely j. A man might bet 2 to i 
against its raining; and if he repeated the offer year by year 
he should come out even in the long run,* provided that he 
can always find someone to take his bet. But another man 
who took into account the information derived from weather 
forecasts could win money off him by accepting the bet only 
in those years when rain was predicted. Ideally then the 
probability on which we should act is the one which is 
relative to all the information obtainable; that is to say, the 
implied class A consists of events which are like a in every 
particular stated. But this only applies when the probabilities 
are known with reasonable certainty. Often they are some- 
what uncertain generalisations based on limited past ex- 
perience. Each additional piece of information cuts down 
the size of the class and thereby makes the generalisation 
more unsafe. Information which we have theoretical reason 
to believe is irrelevant, e.g. whether April 19 does or does 
not fall in Easter week on the occasion in question, should 
be excluded; it only does harm by cutting down the size of 
the class. The question to be settled is then, whether it is 
better to act on a very uncertain probability based on more 
information or a fairly certain probability based on less 
information. This is not the sort of question to be solved 
by mathematics. 

* Subject to a growing fluctuation which the persistent gambler must 
be presumed to have decided to risk. 


Naturally a mathematical theory can take no account of 
the uncertainty of the entities with which it deals, whether 
these entities be probabilities or other numerical quantities. 
By uncertainties I here mean those arising from dubious postu- 
lates, generalisations, etc.; measurable uncertainties, such as 
probable errors, can be (and should be) dealt with mathe- 
matically. The dilemma of having two differently computed 
probabilities to choose from is no different from that which 
arises in regard to many other physical quantities. By one 
method we determine the value of a physical constant with 
very great accuracy, except that there is a doubt whether the 
theory underlying the method is sound; by another method 
we obtain a much less accurate value, but we have more 
confidence in the theory on which it is based. To decide 
which result should have greater weight in determining our 
belief is the kind of job which we have earlier assigned to a 
hypothetical "right-thinking person". The mathematician 
declines to be a candidate for the post. 

The objection to reducing the size of the class and 
thereby making generalisation more unsafe applies to the 
probabilities of everyday life and especially to those based on 
accumulated statistics, but it does not affect probabilities 
(frequencies) which are computed by pure theory. These are 
to be treated as definite probabilities not as merely probable 
probabilities. I do not mean that the theory is certainly true ; 
but it is assumed to be true as the basis of discussion, and it is 
recognised that our results are contingent on the theory being 
right.* The theory will determine the frequency in a narrow 
class as definitely as in a wide class ; there is therefore no dis- 
advantage in cutting down the class, and we incorporate in 
the probability every scrap of information available. 

* For example, one would not compute the probable position of a 
planet on the basis that Einstein's theory has a probability of | and 
Newton's theory a probability of *. If the result is to have any scientific 
usefulness the computer must commit himself to one theory or the other. 


Suppose, for example, we are considering the probability 
that an atom has a velocity within certain limits. We start 
with an initial class commonly supposed to be the class of 
all atoms from which, in the entire absence of information, 
we might suppose our particular atom to have been selected 
at random; the frequency of the given velocity in this class 
is called its a priori probability. I will not stop now to 
discuss this initial class, because it is as it stands a hopelessly 
illogical conception; and a critical study of how it is to be 
placed on a proper footing has very important consequences 
(p. 130). Next a variety of information is to be incorporated. 
The atom is in the earth's atmosphere; it is at a certain 
temperature; it has just undergone a collision with an 
a particle; it is an oxygen atom; and so on. Each piece of 
information as we introduce it cuts down the class by 
eliminating all those members which were inconsistent with 
it. Finally, after each new piece of information has had its 
whack at the diminishing class, we calculate the frequency of 
the given velocity in the class that remains; that then is the 
required probability relative to all known information. 

It is to be noticed that the information is used to define 
what is excluded, not what is included. Events incompatible 
with one of our items of information are excluded; events 
which are consistent with it are not necessarily included, 
because they may be contradicted by another item. This 
Exclusion Method is the only systematic way in which we 
can incorporate a number of separate observational results. 
I think it is. very suggestive of the difference between the 
scientific and the familiar outlook. Ordinarily we expect our 
senses to tell us what there is in the external world; the 
scientist uses them rather to assure himself of what is not 
there. That is to say, he forms as wide a conception of the 
possibilities as he can, and tries to narrow them down by 
crucial experiments. His ideal is to state his conclusions about 
the external world in a sufficiently general form to include 


all possibilities that he is unable to give good reason for 

To illustrate this procedure by exclusion, I recall a question 
once set in a mock examination paper. It is true that it refers 
to the probabilities of everyday life instead of to the definite 
probabilities occurring in scientific theory. But in an ex- 
amination paper the probabilities of everyday life become 
definite for no candidate may doubt information that is 
vouched for by an examiner. The question was 

If A, J5, C, D each speak the truth once in three times (inde- 
pendently), and A affirms that B denies that C declares that D is 
a liar, what is the probability that D was speaking the truth? 

It was many years after I first heard of it that it occurred 
to me that the problem actually had an answer, and moreover 
was an instructive example of the Exclusion Method the 
modification of the a priori probability first stated, by ex- 
cluding those members of the class which are inconsistent 
with the additional information furnished.* The reader will 
be in a better position to appreciate the enormous advantage 
of the exclusion method if he has first been driven wild by 
attempting to solve the problem without it. 

The difficulty in using the exclusion method is to obtain 
a start. The method provides for the addition of knowledge 
to knowledge, but not for the addition of knowledge to 
ignorance. Added information is used to narrow down the 
class of events contemplated; but, starting with complete 
absence of information, how do we obtain the initial class 
to be narrowed down by our first piece of information? In 

* The combinations inconsistent with "A declares, etc." arc truth- 
lie-truth-truth and truth-lie-lie-lie, which occur, respectively, twice and 
eight times out of 81 occasions. Excluding these, D is left with 27-2 
truths to 54 - 8 lies, so that the required probability is 25/71. The solution, 
of course, does not pay heed to the psychology of the quarrel; e.g. we 
do not try to deduce anything from die fact that A was provoked to 
speak rather than to hold his tongue. 


the foregoing problem of A, J3, C and D, our first information 
specified the frequencies of a class so that no difficulty arose; 
but that was due to the benevolence of the examiner. Nature 
does not so kindly adjust her problems to our capacity. So 
the question has often worried us, What class of events 
corresponds to complete ignorance ? The whole conception 
of such a class is a logical contradiction. The height of 
absurdity was reached in the much-discussed Principle of 
Indifference, which asserted that when there is no information 
all alternatives are equally probable. Heaven knows why! 
However, since there are an infinite number of ways of 
classifying alternatives, and the principle does not say which 
way is to be chosen, it leaves us none the wiser. 

There are many instances in which it is plausible to assume 
that a number of alternatives are equally probable. It is not 
always easy to see that the plausibility rests on knowledge 
(or positive conjecture), never on ignorance. The statement 
that the probability is that my next throw will be an ace 
is only true in the sense originally intended if the dice are 
not loaded ; but there is another sense in which the probability 
is still \ even if we suspect that the dice are loaded. The two 
interpretations are due to the class of events not being ex- 
plicitly specified. In the first sense, the probability refers to 
frequency in the class of all throws with this particular cube; 
in the second sense, it refers to all throws with all dice. We 
presume that in the latter class, although loaded dice exist, 
they are loaded against all numbers equally. That is to say, 
we assume that the probability of a certain face of a cube 
bearing a given number, and the probability of its being the 
face nearest to the centre of gravity of the cube, are inde- 
pendent uncorrelated probabilities. This assumption involves 
some knowledge of the methods of making dice. We might 
easily argue against it. If the practice is to stamp the numbers 
in order, so that the number i is on the face which happened 
to be uppermost when the cube was picked up for stamping, 


the tendency will be for the loading to be in favour of the 
number i. The actual practice may be altogether different, 
but my point is that the assumption of equal probability of 
throwing the six numbers is based on information, whether 
true or false, about the circumstances of manufacture. It is 
not true that the probability is | if we have no infoAnarion 
whatever; we must at least know that the usual process of 
manufacture is not that which I have described. 

It will, I think, generally be found that when numerical 
probabilities seem to appear rather mysteriously out of ignor- 
ance, their actual basis is an assumption of non-correlation 
between different frequencies an assumption which, whether 
justified or not by our knowledge of the circumstances, re- 
presents the belief on which we are relying when we assert the 
probability. The belief is positive. It is not adopted merely 
as the most non-committal solution of a problem presented 
by our ignorance. The strength of our belief that the actual 
circumstances are such as not to introduce correlation deter- 
mines the strength of our belief in the consequent probability 


It is notorious that the theory of probability has often been 
applied fallaciously. The most common mistake is to neglect 
the interdependence of two or more probabilities and com- 
bine them by formulae which apply only to independent 
probabilities. As a rule the culprit is fully aware of the 
heinousness of such an offence; it is simply that he has not 
been alert enough to detect the interdependence. Many 
illustrations of this neglect of interdependence could be cited 
from scientific writings up to the present day; but I will 
choose an early example, of which an account is given in 
Bertrand's Calcul des Probabilites. Take warning then from 
the story of Condorcet and the Judges. 
In the first days of its exuberance, the theory of probability 


was applied by some of the famous mathematicians Con- 
dorcet, Poisson, Laplace and others to proposals for mini- 
mising errors of justice. The great Laplace was responsible 
for extravagances scarcely less glaring than those I shall relate. 
The Marquis de Condorcet, who seems to have started the 
idea, was a prominent mathematician of the time. He con- 
sidered the problem of securing that a man should run no 
more risk of being wrongfully condemned than he might be 
expected willingly to shoulder. Take, for instance, the pro- 
portion of those who were accidentally drowned at a certain 
crossing of the Rhone to the number who safely passed it. 
No one troubled about this risk; therefore they would cheer- 
fully accept the same risk of being executed by mistake. By 
such considerations Condorcet decided that one miscarriage 
of justice in 144,768 trials was a suitable figure to aim at. He 
had assured himself that truly enlightened judges could be 
found who would deliver not more than one wrong judg- 
ment in five. Here the wonderful new theory came in. Take 
sixty-five such judges and require a majority of nine for a 
judgment against the prisoner, and the risk of a wrong 
sentence is reduced to the above figure. 

It does not seem to have occurred to Condorcet that the 
truly enlightened judges might be to some extent guided by 
the evidence ; and that when an innocent man is wrongly con- 
demned it is usually because the evidence has seemed to point 
against him. His calculation had assumed that the right and 
wrong decisions of his sixty-five judges would be distributed 
independently of one another. 

Condorcet was somewhat concerned lest, with the large 
increase of judicial posts, there might be insufficient judges 
of the same high standard. Still judges who made, say, one 
mistake in three could be used; it was only necessary further 
to increase their number. His only misgiving was that if, 
other classes being exhausted, it was necessary to include 
those who made more than one mistake in two, his method 


would break down. ("Not at all", adds Bertrand. "A 
sufficiently numerous assembly in which each member is 
wrong more often than not will certainly pronounce against 
the truth, and therefore give a sure means of knowing it at 
least according to Condorcet's formulae.") 

Eight years later the Revolution came. Well had it been 
for the Marquis de Condorcet could he have been assured of 
one truly enlightened judge. He died by his own hand to 
escape the tribunal. 


Another source of fallacy is inverse probability or the prob- 
ability of causes. In science the ' ' causes " are usually alternative 
hypotheses or explanations. It is argued that if a certain 
observed result is 100,000 times more probable on hypothesis 
A than on hypothesis J3, then hypothesis A is 100,000 times 
more probable than hypothesis B. In judging the credibility 
of the two hypotheses we should, of course, regard informa- 
tion of this kind as highly pertinent; but there is no justifica- 
tion for the inverse form of statement. Suppose that you 
take a penny from your pocket and, tossing it five times, 
note that it turns up heads each time. The chance of a sequence 
of five alike throws with a normal penny is 7^ ; with a 
double-headed penny the chance is unity. But you would 
not argue that it is 1 6 times more probable that your penny 
is double-headed than that it is normal, 

We must not, however, forget that probability is always 
relative to the information supplied. In rejecting the argu- 
ment that the penny is most probably double-headed, we use 
our secret information that double-headed pennies are rare. 
Setting aside that and all other information which is not 
openly stated, we should have had no particular reason to 
reject the probability of 16 to i as being the probability 
relative to the given information. On the other hand there 
is no reason to accept it* When the argument is examined in 


detail, it is found to assume that, prior to the tossing, it was 
equally likely that the penny we had got hold of was double- 
headed or normal. Even if this were true, it is just as ille- 
gitimate for the defender of inverse probability to use his 
secret information to this effect as it is for us to use our secret 
information to the contrary. The deduction of the probability 
of causes from the probability of their consequences is a game 
whose rules are such that no one can take part in it without 

But how are we to get on in physics without inverse 
probability? All our knowledge of the external world is an 
inverse inference an inference of cause from effect. We 
experience sensations and we attribute them to more or less 
probable causes existing in the external world. Let it first be 
said that as regards the general scheme of physical law in- 
ferred from our experience the accepted key to the crypto- 
gram we do not attribute to it any numerical probability. 
The evidence appears to us to warrant a strong belief in it, 
and that is all we can say. But as regards observed individual 
features, we commonly state our conclusions as numerical 
probabilities. We measure the parallax of the star Capella, 
and infer a probability of - 6 that Capella is between 13 and 16 
parsecs away from us. This is really a statement of inverse 
probability, for the actual calculation is that the set of 
measurements that we have made is one which was (before 
we made it) five times more likely to occur if Capella is 
within these limits of distance than if it is outside them. 

I think that there is a more logical way of expressing our 
detailed knowledge of the universe. The science of astronomy 
will not collapse if it turns out that we have made a wrong 
inference about Capella. We can never be sure of particular 
inferences; therefore we should aim at a system of inference 
that will give conclusions of which in the long run not more 
than a stated proportion, say i/q, will be wrong. Hence, 
instead of making the definite inference that Capella is 


probably between 13 and 16 parsecs away (probability |), 
we make the probable inference (probability |) that CapeUa 
is definitely between 13 and 16 parsecs away. By adopting 
the latter form we use a direct instead of an inverse prob- 
ability and the logical difficulties of the former form are 

Accepted observational knowledge of the universe is then 
a function of q a series of maps becoming more and more 
detailed as q decreases. Thus in the map in which ^ of the 
features are correct, we place Capclla between 12 and i8 
parsecs away. In the map in which - 6 of the features are 
correct, we place it between 13 and 16 parsecs; we can afford 
to be more precise in our statements as we become more 
reckless of their truth. The series of maps starts (at ^infinity) 
with a map which is entirely correct but unfortunately 
entirely blank; it ends (at q=i) with a map full of the 
minutest detail of which only an infinitesimal proportion is 
correct. What a philosopher is to make of these maps I will 
not venture to say ; but the scientist affirms that some of the 
intermediate maps (say between q=$ and q=^2o) can be of 
considerable assistance to a sojourner in the universe who has 
to find his way about. 

To see how this outlook avoids any question of inverse 
probability, we may refer again to the double-headed penny. 
You have tossed up the penny five times and it has fallen 
heads (or alternatively tails) every time. It is suggested that 
you should infer that the chances are 15 to i that it is a 
double-headed (or double-tailed) penny. That is clearly not 
the right inference. But suppose that you do not claim to 
be making the "right" inference, but to be applying a system 
of inference of facts (not of chances which would be meaning- 
less in the circumstances) which will lead you wrong not 
more than once in 16 times. You may then boldly infer that 
the penny which has fallen heads or tails five times in 
succession is abnormal. We have secret knowledge that you 


will be wrong this time. But you will only be wrong a 
regards one penny out of every 16 that you try; for 15 ou 
of 16 will assure you of their normality by exhibiting botl 
faces. Thus your system of inference fulfils what you clairr 
for it. 


Probability has intertwined itself round the roots of physica 
science. In thermodynamics, in quantum theory, and when- 
ever gross matter is treated as an aggregation of a vast numbei 
of particles, the laws of chance are involved. Probability 
leavens the secondary scheme of physical law the laws 
which are obeyed because it is "too improbable" that they 
should be broken. This application demands our specia] 

We have had examples of two ways of utilising an obser- 
vation. We can consider what may be inferred from that 
observation alone and calculate the probability attached to 
our inference; or we can consider how the new information 
contained in the observation modifies the probabilities which 
corresponded to the previous state of our knowledge. The 
second is the exclusion method discussed in Section m; it 
lends itself to more systematic treatment and is used through- 
out thermodynamics and quantum theory. In the modern 
form of quantum theory, known as wave mechanics, the 
exclusion method has been developed into a fine art. Each 
observation is treated as excluding a number of alternatives 
which had not been inconsistent with earlier knowledge and 
were accordingly represented as existing in the probability 
distribution or "fog" whose history is being traced. 

Broadly speaking wave mechanics pictures a universe 
whose substance is probability, whereas classical mechanics 
pictures a universe whose substance is mass, energy, mo- 
mentum, electric and magnetic force, etc. In wave mechanics 
we examine the way the probability moves about and re- 


distributes itself; in ordinary mechanics we find the way 
mass, momentum and electromagnetic field move or are 
propagated. In the former the waves, which give the subject 
its name, are waves of probability; in the latter we treat 
sound waves, electromagnetic waves and gravitational waves. 
For brevity these may be contrasted as a universe of prob- 
ability and a universe of entities. They are, however, both 
aspects of the same universe whose description involves both 
probabilities and entities. The difference in point of view is 
that in the first we attach entities (electrons, protons, photons) 
to the probabilities which we study; in the second we attach 
probabilities to the entities which we study only from the 
nature of the entities treated in classical physics the attached 
probabilities are all practical certainties (p. 78). In macro- 
scopic physics the variety lies in the entities greater or lesser 
masses, greater or lesser field strength the probabilities 
being all similar units; in microscopic physics the position 
is inverted and the variety lies in the probabilities, the entities 
generally being all similar units, e.g. electrons. It is therefore 
found to be more businesslike and practical to contemplate 
a distribution of probabilities ; and the entities attached to 
them tend rather to drop out of sight in our calculations and 

We have seen that in order to use the exclusion method 
it is necessary to start with an initial class ; and wave mechanics 
accordingly starts with an initial or "a priori probability 
distribution" of the positions and velocities of the electrons 
or other entities. A priori probability is essentially an un- 
observable, for when we introduce observational knowledge 
we obtain a modified probability relative to that knowledge. 
We therefore seem led into the old fallacy of the principle 
of indifference in supposing that there can exist a probability 
relative to complete ignorance. This is a most unsatisfactory 
feature of wave mechanics when considered by itself; but the 
difficulty disappears when wave mechanics is combined with 


relativity theory. The a priori probability distribution is then 
regarded in the same way as other unobservables are regarded 
in relativity theory, e.g. a frame of space and time. For the 
purpose of representation we adopt an arbitrarily chosen 
frame of space and time; but our choice makes no difference 
in the end when we translate our results directly into terms 
of what can be observed. Similarly we can adopt an 
arbitrarily chosen distribution of initial probability; our 
choice makes no difference in the end when we translate our 
results directly into terms of what can be observed. 

Thus the initial probability referred to in quantum theory 
and in the kinetic theory of gases is not an a priori probability 
in any metaphysical sense. It is part of an arbitrarily chosen 
reference system; and it is no more necessary to decide 
whether one distribution of initial probability rather than 
another is the true inference from complete ignorance than 
it is to decide whether the yard or the metre is the true 
standard of length. We adopt any convenient a priori 
probability distribution as we adopt any convenient frame 
of space and time; but it follows from the unobservable 
character of these comparison systems that the laws of physics 
must be invariant for all transformations of them. The re- 
cognition of this invariance is another of the important steps 
in the unification of relativity theory and quantum theory.* 

When you make a change of your system of reference, 
whether it be a change of the frame of space and time or of 
the initial probability distribution to which all observational 
information is applied, you must carry through the change 
to the bitter end. If you change your space-time frame in 
mechanics you must change it also in optics, otherwise you 
will reach erroneous conclusions in regard to an experiment, 
such as the Michelson-Morley experiment, in which both 
optics and mechanics are involved. Just as in former days the 
Michelson-Morley experiment was misunderstood through 
* Previous steps are discussed on pp. 48, 108. 


segregating the optical and the mechanical (or metrical) 
factors in the experiment, so at the present time our experi- 
ments on atoms and electrons are very generally misunder- 
stood through segregating the microscopic (quantum theory) 
and macroscopic (relativity theory) factors in the experiment. 
In particular if, in considering an experiment on an electron, 
you change the adopted a priori probability distribution of 
position and velocity, you must consider the consequences of 
that change not only on the formulae describing the be- 
haviour of the electron itself but on all the particles that make 
up the apparatus used in the experiment. For the result of 
the experiment is affected just as much by a change of 
behaviour of the apparatus as by a change of behaviour of 
the electron. 

Since physics has been divided into two branches, quantum 
theory and relativity theory, the electron being studied by 
the former and the gross matter of the apparatus by the latter, 
the experiment has been under the charge of two partners 
neither of whom knows what the other is doing. Relativity, 
dealing with matter and field on the gross scale, treats of the 
averages associated with vast numbers of particles. Leaving 
aside electrical characteristics, it treats especially of the average 
energy of the particles and the associated quantities, average 
momentum and average stress-system; these are grouped 
together to form what is called an average energy-tensor. 
So the a priori probability distribution in quantum theory is 
represented in relativity theory by its average energy-tensor. 
But when it enters into relativity theory it receives a new 
name; it is called the fundamental or metrical tensor (g^ v ). 
This is the characteristic of space (or aether) which deter- 
mines what will be the measure of the distance of two 
specified points or of the interval of time between two 
specified events. 

Now let us return to the quantum theory side of the 
partnership. The quantum physicist is studying, let us say, a 


system of two or three electrons whose positions he has 
temporarily denoted by certain symbols called coordinates. 
But he cannot tell what these symbols mean in an obser- 
vational sense he cannot tell what are the distances between 
the particles without appealing to his partner to furnish him 
with the metrical tensor g^ v which constitutes the code for 
translating the symbols into distances. Just as in an actual 
experiment he would have to borrow gross apparatus be- 
longing to macroscopic physics to measure the distances, so 
in the theory he has to borrow a macroscopic tensor to 
calculate them. So he borrows the tensor g^ v from the 
relativity physicist. What he generally fails to recognise is 
that this is simply the averaged characteristics of his own a priori 
probability distribution being handed back to him. 

Unconscious of this identity the quantum physicist applies 
the metric g** v to the positions and directions of motion of his 
particles and hence introduces it into his description of their 
probability distribution. In particular, it is with reference to 
this metric that he describes the initial a priori probability 
distribution the framework into which observational know- 
ledge is incorporated by the exclusion method. He discovers 
a remarkable result! He finds that the initial probability 
distribution must be uniform and isotropic throughout space- 
time. This is not very surprising when we recall that the 
initial probability distribution furnishes the metric g* which 
is then employed to measure the initial distribution. 

I imagine him turning on me and saying "You were 
wrong when you said that I was free to choose the initial 
a priori probability distribution arbitrarily. Nature has chosen 
a uniform and isotropic distribution, and forces her choice 
on me. Any other choice would not lead to the equations 
which are verified by experiment". I might reply by re- 
minding him, that by the way in which it is used in connection 
with observational knowledge the initial distribution is neces- 
sarily outside observation, so that there must be a fallacy in 


his conclusion. But the important thing is to see the source 
of the fallacy. By using g^ v for the metric in his description, 
he is using the initial probability distribution to describe the 
initial probability distribution. However arbitrary it may be 
by extraneous standards, compared with itself it is necessarily 
exhibited as uniform and isotropic. 

Those apparent laws of Nature which express uniformity 
and isotropy arise because we measure the world with 
apparatus which is itself part of the world. The measuring 
apparatus and that which is measured are constituted ulti- 
mately of the same type of elementary particles; so that any 
asymmetry of behaviour must appear on both sides of the 
comparison and be eliminated in all our measurements. 
I have explained elsewhere* how Einstein's law of gravita- 
tion, which states that the curvature of the world in empty 
space or aether is uniform and isotropic, arises in this way. 
The uniformity and symmetry of the a priori probability 
distribution is of similar character, and is in fact a closely 
related aspect of the same investigation. 

Coming back to the general theory of probability which 
is the subject of this chapter, we have been concerned to show 
that probability is always relative to knowledge (actual or 
presumed) and that there is no a priori probability of things 
in a metaphysical sense, i.e. a probability relative to complete 
ignorance. We have examined what at first appeared to be 
two cases of exception. In the example of the loaded dice, 
we have pointed out that what is assumed is not ignorance 
but knowledge that the circumstances of manufacture are 
such that two probabilities concerned in the problem are 
uncorrelated. The other example is the initial probability 
assumed in wave mechanics and other statistical branches of 
physics, which is commonly called a priori probability. It is, 
for example, assumed that the initial probability of finding 
a particle in a given region is simply proportional to the 
* The Nature of the Physical World, pp. 138-145. 


volume of die region; in other words all equal volumes have 
an equal amount of initial probability of containing the 
particle. This is often looked upon as an example of the 
Principle of Indifference that initially (i.e. when no infor- 
mation is supplied) all alternatives are equally probable. But 
it has nothing to do with that principle. The proportionality 
of volume to initial probability is a physical law of precisely 
the same type as the proportionality of energy to mass. Such 
laws arise because there are two ways in which the same 
natural entity can affect our experience. Except that they are 
measured in different units mass is simply an alias of energy. 
Similarly the volume of a region is an alias of the initial 
probability of its containing the particle, the one name being 
used in macroscopic theory and the other in microscopic 


Study is like the heaven's glorious sun, 

That will not be deep-searched with saucy looks. 

SHAKESPEARE, Loves Labour's Lost. 


THE history of exploration of the interior of a star begins in 
the year 1869 when J. Homer Lane wrote a famous paper 
entitled "On the Theoretical Temperature of the Sun, under 
the Hypothesis of a Gaseous Mass maintaining its Volume by 
its Internal Heat, and depending on the Laws of Gases as 
known to Terrestrial Experiment". He might perhaps have 
chosen a more snappy tide. But the fullness has the advantage 
of bringing before us a number of important ideas. The 
various phrases each deserve close attention, and we shall use 
them as the firstly, secondly, thirdly, of our sermon. We 
shall consider other stars besides the sun, and other conditions 
of the interior besides the temperature ; but everything centres 
on the problem of temperature. What is the degree of heat 
deep down inside these great celestial furnaces? 

I would emphasise the phrase "depending on the laws of 
gases as known to terrestrial experiment". There is no specu- 
lative intention in these studies of the interior of a star. We 
simply want to find out how far the phenomena which we 
observe in the sky agree with and are a consequence of the 
laws that have been assigned to matter as the result of 
laboratory experiment. We encounter matter under con- 
ditions very different from those of the laboratory; and it 
may have something fresh to tell us something quite un- 
foreseen. But anything essentially new has to be sorted out 


from that which is a direct consequence of what we already 
know, or think we know. If the stars have any revolutionary 
ideas to suggest they will show it by a discordance from the 
results which we calculate for them on the basis of the 
accepted laws of physics. 

Before diving into the interior, I must refer to our general 
knowledge of the stars as seen from outside. The number of 
stars within range of our most powerful telescopes is of the 
order of a thousand million, or say one apiece for every 
inhabitant of the earth. These, and many more stars too faint 
to be detected, form a great system which we call the Galaxy. 
This system is not the whole universe; but what lies beyond 
it will occupy us in Chapter x. The stars show a very wide 
diversity. Some are extremely dense and compact, others 
extremely tenuous. Some give out a million times as much 
light and heat as others. Some have a surface-temperature as 
high as 20,000 or perhaps 30,000, others not more than 
3000. Some stars are believed to be pulsating, swelling and 
deflating with a period of a few hours or days. A considerable 
proportion occur in pairs two stars revolving round each 
other. Some flock in clusters; the members of such clusters 
though widely separated from each other have at least the 
connection of a common origin. One star, we know, has a 
system of planets, and from one of those planets we view 
it; whether any other stars have such a system can only be 
guessed. According to Jeans there is theoretical reason to 
suppose that the evolution of a planetary system is a rare 

The most uniform characteristic of the stars is their mass, 
that is to say the amount of matter which constitutes them. 
A range from f to 10 times the mass of the sun would cover 
all but the most exceptional objects. The general run of the 
masses is within a much narrower range. The most massive 
stars tend to force themselves on our notice because they are 
the most luminous; if we eliminate this selective effect, the 


diversity of mass among a hundred stars picked at random 
would probably be not much greater than among a hundred 
men, women and children picked at random from a crowd. 

When the spectroscope was applied to the detection of the 
various elements in the heavenly bodies, the first impression 
was that the stars varied greatly in chemical constitution. 
Elements prominent in the spectrum of one star are absent 
in another. Some stars, such as Sirius, show a spectrum 
which is almost wholly hydrogen; in the sun iron is very 
prominent; among the cooler stars, in which chemical com- 
pounds are not wholly dissociated by temperature, some 
indicate carbon compounds and others rather oddly specialise 
on titanium oxide. But these are not real differences of 
chemical composition. A particular spectrum can only 
appear if the physical conditions are such as are required to 
stimulate it. The variety of stellar spectra is therefore due 
primarily to the variety of physical conditions differences 
of temperature and pressure in the layers which the spectro- 
scope explores. We cannot be sure that the stars all have the 
same chemical composition; but if there are differences, it is 
by no means a straightforward problem to ascertain them. 

In any case the composition of the layers bubbling on the 
outside of the stellar furnaces cannot be taken as a safe guide 
to the composition within. So we start our investigation of 
the stellar interior in practically complete ignorance of its 
chemical constitution. Leaving aside possible differences of 
chemical constitution, the stars may be expected to form a 
twofold sequence. We may specify a star by its mass and 
radius the total amount of matter, and the space into which 
it is packed. We anticipate that these two data will fix all 
the other characteristics of the star how much light and 
heat will pour out of it, what temperature its surface will 
take up, what will be the period of its pulsation if pulsation 
is possible, and so on. These presumably are necessary pro- 
perties of a given amount of material forming a globe of 


given size, and it ought to be possible to calculate them by 
a study of the physical conditions. At any rate that is the 
line on which we may start to work, and if there are additional 
complications they will appear in due course. 

It would be as difficult to select a "typical star" as to select 
a typical animal to represent the animal kingdom. But the 
sun is about as typical as any. It is not at all extreme in any 
of its characteristics; and around us there are numerous stars 
which are practically replicas of the sun. The sun is 330,000 
times greater than the earth in mass and 1,300,000 times 
greater in volume. Its diameter is 865,000 miles, and its mass 
is 2000 quadrillion (2 . io 2 ?) tons. Its mean density is rather 
greater than that of water. 


Viewing the sun from outside, we look down through the 
semi-transparent outermost layers. The level which is roughly 
the limit to which we can see down is called the photosphere. 
It is ascertained by fairly direct observational methods that 
the temperature at that level is nearly 6000 Centigrade. 
Continuing inwards below the photosphere the temperature 
must become higher and higher until it reaches its maximum 
at the centre of the sun; but we can only follow this increase 
by theory. It is found that by far the greater part of the 
interior mass is at a temperature above a million degrees. 
According to a favourite mathematical model the sun's 
central temperature is 21,000,000 and the mean temperature 
of the whole mass is 12,000,000. These figures probably err 
in being, if anything, too high. 

The clue which we follow in finding the internal tem- 
perature is contained in the title of Lane's paper "a gaseous 
mass maintaining its volume". The mass of 2. io 2 ? tons which 
constitutes the sun must exercise enormous internal pressure. 
If it were devoid of heat the matter would be crushed by this 


pressure into that strange condition which we find in the 
Companion of Sirius and other white dwarf stars, where the 
density is thousands of times greater than that of any material 
known on earth. Heat is required to distend the matter so 
as to occupy the actual volume of the sun. By heat we mean 
the energy of the random motions of the molecules. If the 
planets were deprived of motion they would all fall into the 
sun; so we may say that the solar system is kept distended 
by the motions of the planets. In the same way the sun is 
kept distended by the heat motions of the particles of which 
it is composed. By bringing together the various physical 
laws which bear on the subject, we have been able to make 
a tolerably close calculation of the amount of heat required 
to give the observed distension; and also to determine, but 
more roughly, how the heat must distribute itself through 
the sun in order to preserve a steady state. 

The very high temperature has one effect which was not 
at first realised. There are two forms of heat material heat, 
which is the energy of the particles, and radiant heat, which 
is the energy of aether waves. At terrestrial temperatures, 
for example in a white-hot mass of metal, the radiant heat 
is quite insignificant compared with the material heat. If we 
go near the white-hot mass we feel a great deal of radiant 
heat coming from it, but this is produced at the moment of 
emission by converting material heat in the iron; it is manu- 
factured as required, and practically no reserve stock is kept. 
When the temperature is increased, the material heat increases 
roughly in proportion to the absolute temperature, but the 
radiant heat contained in the body goes up as the fourth 
power of the temperature, so that it gradually overtakes the 
material heat. Even at the temperature of the sun there is 
not so much radiant heat as material heat; and except possibly 
in a few of the most massive stars, the advantage is always 
with the material heat. But there is no longer any great 


I have little doubt that it was this approximate balancing 
of the two forms of heat at some early stage of the evolution 
of a star that determined the standard mass to which the stars 
more or less closely conform. Something must have decided 
that the matter constituting our Galaxy has not all con- 
densed into one mass but has divided into thousands of 
millions of stars, the majority of which are surprisingly 
uniform in the amount of material they contain. The mass 
of the pattern star cannot have been arbitrary. It seems sig- 
nificant that the mass is such that (especially in the earlier 
stages of condensation) the radiant heat is nearly on a parity 
with the material heat. For 50-fold greater or 5O-fold smaller 
mass we should not have anything approaching a balance. 

In Lane's discussion, and for a long time afterwards, the 
existence of this large quantity of radiant heat was not re- 
cognised. When the heat of the star was thought of as wholly 
material, it was necessary to postulate some means for bringing 
it up from the interior to the surface where heat is being 
radiated away. It was supposed that the matter at the surface 
cooled and sank down, and hot matter from below came up 
to take its place; so that throughout the sun there were 
convection currents bodily transferring the heat to the points 
where it had been lost and incidentally keeping the material 
well-stirred. But now the boot is on the other leg. Ever 
since it was recognised that the stars contain a vast quantity 
of radiant energy, the problem has been, not to devise a way 
of bringing heat up to the surface, but to understand how 
this highly mobile form of energy is held back from the 
surface how it is encaged by the matter so that it does not 
leak out faster than we observe it to do. 

At a temperature of some millions of degrees radiant energy 
:onsists of X rays. So in the stellar interior we have X rays 
n great abundance travelling in all directions. If the atoms 
md electrons in the sun were suddenly abolished the X rays 
low confined in the interior would scatter through space 


with the speed of light; 300,000 years' supply of radiation 
would be squandered in an instant. The atoms dam back this 
flood, catching and turning away the aether waves as they 
try to escape, absorbing and re-emitting them in a new 
direction, so that they go aimlessly round and round the 
maze. Thus only a slight leakage of radiation dribbles out to 
illuminate and warm the earth and the other planets. 

This leads us to the principal aim of investigation of the 
stellar interior. Having found the internal distribution of 
temperature in the star and knowing therefore the quantity 
and quality of the radiant energy imprisoned there (for this 
is determined solely by the temperature), knowing also the 
density of die matter and therefore the number of atoms 
engaged in holding back the radiant energy, we ought to be 
able to calculate how much escapes. It is like calculating the 
flow of water through a pipe, when we know the head of 
water causing the flow and the resistance obstructing the 
flow. Here the increasing concentration of the X rays as the 
temperature increases inwards supplies the pressure gradient, 
and the opacity of the matter obstructing the transmission 
of X rays supplies the resistance. Our aim then is to calculate 
from the known laws of absorption of X rays how much 
radiation will get through, and so ascertain theoretically the 
amount of the energy flow which is slowly leaking outwards 
through the star. If our calculation is right, it will agree with 
the amount which emerges at the surface and constitutes the 
light and heat of the star. The calculation therefore gives us 
the brightness of the star, or more strictly the "heat-bright- 
ness" which measures the total radiant energy emitted irre- 
spective of its luminous efficiency. The heat coming to us 
from many of the stars has been measured directly; but in 
any case the heat-magnitude of a star can easily be computed 
from its light-magnitude by applying a well-known correc- 
tion depending on the colour or spectral type. 

Effectively therefore by the method here outlined we 


ought to be able to compute from the mass and radius of a 
star its theoretical luminosity. We can then compare this 
calculated luminosity with the luminosity observed. Before 
describing the results of this comparison there are a number 
of points to be considered. 

As the escaping radiation travels from the hot interior to 
the comparatively cool surface layers, it is gradually trans- 
formed from X rays to longer wave-lengths or lower fre- 
quencies; so that the radiation which finally emerges consists 
of visual light together with some ultra-violet and infra-red 
rays as shown in the star's spectrum. The stepping-down of 
the frequency is automatic. Each unit of radiant energy 
each photon is being absorbed and re-emitted every few 
inches on its journey outwards; so that there is ample 
opportunity for adjusting the composition of the radiation 
to that proper to the temperature of the region which is 
being traversed. If we were to follow the last stages of the 
journey we should be concerned with the absorption of light 
instead of the absorption of X rays. But fortunately there 
is no need to trouble about this. Having, as it were, con- 
ducted the outflowing stream through ^ of the material 
of the star, we can leave it to find its own way out. I say 
fortunately, because it is much easier to treat temperatures 
above a million degrees. It is the high temperature of the 
stars which makes our problem soluble. We could no doubt 
treat cooler matter in a similar way if we were given the 
necessary data. But we are not in a laboratory where we 
can find out any data we require; we are in the interior of 
a star provided with next to no data. 

In this part of our discussion we are not concerned with 
the ultimate source of a star's heat. We take the star in the 
condition in which it now is and calculate that radiation is 
flowing through and out of it at a certain rate. Clearly there 
will be a gradual change in the condition of the star unless 
the heat inside it is being replenished from some source in 


the interior; and we infer that such replenishment occurs, 
because without it the star would change too rapidly for any 
admissible time-scale of evolution. But "rapid" here means 
perceptible in a thousand or a million years; it is not the kind 
of unsteadiness which would upset our calculation. So far as 
the calculation of the luminosity is concerned we do not care 
whether the star's store of heat is being replenished or not. 

There is just one point at which our problem is not 
entirely detached from the problem of the source of main- 
tenance of a star's heat. To obtain an exact result we should 
have to know how the source is distributed through the 
star whether it is concentrated in the hottest central regions 
or is evenly distributed through the mass. We meet this 
difficulty by considering both extremes of distribution in 
turn, and calculating the luminosity on both hypotheses; the 
truth must lie between them. In this way it is found that 
the extreme uncertainty arising from our ignorance of the 
distribution of the source is for a typical star no more than 
o m *5. We shall be well-satisfied if our calculation reaches 
this order of accuracy. 

There can be little doubt that the heat of the sun and of 
other stars is being maintained by the liberation of some 
form of subatomic energy in the interior. Until the last two 
or three years the laboratory physicist had no information as 
to the conditions of release of subatomic energy; so on this 
side of the subject the astronomer could get no help from 
physics. Thus in developing the theory of the constitution 
of the stars depending on the laws of physics "as known to 
terrestrial experiment" progress was contingent on our 
being able to separate off an independent field of research 
which did not involve the unknown laws of subatomic 
energy. The problems treated in this chapter are segregated 
from the problem of subatomic energy in this way except 
to the insignificant extent referred to in the last paragraph. 
Circumstances are now changing, and a great number of 


processes which release subatomic energy are being studied 
in the laboratory. It may be expected that before long 
important developments in the application to the source of 
stellar energy will follow. Some account of this side of the 
problem of stellar equilibrium will be given in Chapter vni. 


We have seen that an atom consists of a heavy nucleus 
together with a number of loosely attached satellite electrons 
belonging to it as the planets belong to the sun. By various 
kinds of maltreatment the physicist is able to detach one or 
more of the satellite electrons; this process of chipping off 
electrons is called ionisation, and the mutilated atom is called 
an ion. In laboratory conditions there is not much difficulty 
in producing a few ions among a great number of normal 
atoms; but when a certain proportion have been ionised, so 
that there are many homeless electrons wandering about, the 
ions are continually capturing these vagrants, and we soon 
reach a stage at which atoms are being made whole again as 
fast as we can ionise them. 

At a temperature of 10 million degrees the forces of dis- 
ruption are enormously intensified. Ionisation instead of 
being an occasional disease is epidemic. The intensification is 
in quantity rather than in quality. In a contest between the 
sun and the Cavendish Laboratory as to which could do the 
most violence to a single atom, I would back the Cavendish 
Laboratory. For the purpose of electrical experiments there 
is abundance of energy on the sun but very poor insulation. 
The efficiency of the sun is in mass-production. That helps 
us in our problem; for if mass-production is the only new 
feature, the multiplication table suffices to cope with it. 

For definiteness consider an iron atom in the deep interior 
of the sun. Normally it should have 26 satellite electrons; 
22 of these have broken loose and are wandering freely 


through the material. Lighter elements such as carbon are 
stripped bare to the nucleus. Wandering electrons are always 
trying to settle in the vacant orbits and may succeed for an 
instant; but immediately an X ray comes along and explodes 
the electron away again. Perhaps I should add that the 
electron "wanders" at an average speed of 10,000 miles a 

You can picture the commotion at 10 million degrees in 
the interior of the sun. Crowded together within a cubic 
centimetre there are more than a quadrillion (io* 4 ) atoms, 
about twice as many electrons, and 20,600 trillion X rays.* 
We can speak of the number of X rays for the quantum theory 
gives them a kind of atomicity; each of them is a "photon" 
capable of exploding a satellite electron from an atom. The 
X rays are travelling at 186,000 miles a second (the speed of 
light) and the electrons, as already stated, at 10,000 miles a 
second. Most of the atoms are hydrogen atoms or rather, 
since they have lost their satellite electrons, they are simply 
hydrogen nuclei or protons; these are travelling at 300 miles 
a second. Here and there we find heavier atoms, such as iron, 
lumbering along at 40 miles a second. Now you know the 
speeds and the state of congestion of the road; and I will 
leave you to imagine the collisions. It is not surprising that 
the atoms are found with their garb of electrons somewhat 
torn or even stripped naked. 

These motions are those already referred to as distending 
the sun. We assign internal temperatures such that the corre- 
sponding motions are just sufficient to keep the sun distended 
to its observed volume. The calculation involves many 
technicalities which need not be mentioned here; but one 
datum is essential, viz. the average weight of the freely 
moving particles. The higher the average weight, the higher 
is the deduced temperature. The results are rather sensitive 

* In this subject the opportunity of giving a number correct to I per 
cent, occurs so rarely that I have fallen to the temptation. 

ENPS 10 


to this, so that unless the average weight is known fairly 
accurately the computed internal temperature, and more 
especially the computed luminosity, may be badly out. But 
how can we decide the average weight of the particles, 
being, as we are, ignorant of the chemical composition of the 
material in the interior? Here the ionisation of the atoms 
plays a very important part. We can best show this by a table 
Sfor a representative selection of elements. 


No of 

of atom 

weight of 







For example, if the sun were made entirely of oxygen and 
there were no ionisation, the average weight of the particles 
would be the atomic weight of oxygen (16) ; but ionisation 
splits each atom into 9 particles 8 electrons and a nucleus 
and the average weight is therefore 16/9 or 1-78. The im- 
portant feature is the steadiness of the average weight given 
in the last column of the table. It does not matter what 
element we choose from lithium onwards, or what mixture 
of elements; the average weight is always in the neighbour- 
hood of 2. How different would it have been without 
ionisation! We should then have had a possible range from 
7 to 197 instead of a possible range from 1*7 to 2*5; and we 
could not have made much progress without knowing de- 
finitely the composition of the star. 

It would almost seem that Nature has taken a special 
interest in smoothing away our difficulties, for the small pro- 
gression in the last column of the table is actually beneficial. 


If by any chance the sun is made of gold its internal tem- 
perature is substantially higher than if it is made of oxygen 
(owing to the difference of 1-78 and 2*46). But we are not 
so much interested in the precise value of the internal tem- 
perature as in the flow of radiation through the star which 
results from it. Mass for mass, gold offers more obstruction 
to the passage of X rays than oxygen does, and this just about 
counteracts the effect of the higher temperature. In short, 
when we calculate the brightness of a star from its mass and 
radius we obtain practically the same result whether the 
material be oxygen or gold or any other element within the 
limits of the foregoing table. 

But hydrogen is an exception. The hydrogen atom of 
weight i is broken into two particles, a proton and an 
electron, so that the average weight is . This is too large a 
deviation from the normal value 2 to be ignored. If a large 
proportion of the material is hydrogen the internal tem- 
perature is substantially lowered and the calculated brightness 
is reduced very considerably. Broadly speaking we need 
distinguish only two kinds of matter inside a star, namely 
hydrogen and not-hydrogen. 

I think that the one important change in the last seven 
years in the theory of the stellar interior is the recognition 
that hydrogen is very abundant.* You will find, for example, 
that in my book Stars and Atoms (1927) the conclusions are 
given subject to the reservation that there is not an excessive 
proportion of hydrogen (pp. 22, 36). We now believe that 
this proviso is not fulfilled. Ten years ago it was known that 
(on the usual assumption that the material was not-hydrogen) 
the calculated luminosities of the stars came out systematically 
too bright, and that this discrepancy could be cured by 
admitting sufficient abundance of hydrogen. Perhaps it will 

* B. Stromgren, Zeits.fur Astrophysik, vol. 4, p. 118 (March 1932); 
A. S. Eddington, Monthly Notices of the R.A.S., vol. 92, p. 471 (April 



be thought that the hydrogen explanation of the discrepancy 
might have been adopted then. But soon afterwards atomic 
physics was in the throes of a revolution, the older theory 
being replaced by wave mechanics; and until the laws of 
absorption of X rays were re-investigated on the new theory 
it was uncertain whether the discordance might not originate 
there. Gradually all loopholes seem to have been closed up, 
and we are apparently driven to adopt the hydrogen ex- 
planation. Simultaneously hydrogen has been found to be 
excessively abundant at the outside of a star, according to the 
interpretation of the spectroscopic observations. 

There was another necessary step in the argument. 
Hydrogen, as the lightest of the elements, might be expected 
to diffuse to the outer part of a star; if so, it would not play 
the part we want it to play in lowering the temperature of 
the deep interior. Until this objection was met, it could not 
be claimed that abundance of hydrogen would cure the 
discrepancy between the calculated and observed luminosities. 
We now find that the slow diffusion of hydrogen to the 
surface is counteracted by a stirring of the material. The 
early theory of convection currents in the interior has been 
abandoned; but it is found that the rotation of the star must 
give rise to an up and down circulation which, although 
exceedingly slow, stirs the material faster than the hydrogen 
can work its way to the surface. The star may therefore be 
assumed to be of uniform composition throughout almost 
the whole interior.* 

It seems possible now to make a reasonably trustworthy 
determination of the amount of hydrogen in a star if we 
know its mass and luminosity and, very roughly, its radius. 
We find that there is only one way in which such a mass 

* The argument is not intended to apply to the outermost layers where 
the conditions are very different from those typical of the interior. 
Accordingly we must not assume that the uniform internal composition 
is the same as that found by spectroscopic examination of the surface. 



could give the observed amount of light, namely it must 
consist of 30 per cent, hydrogen and 70 per cent, not- 
hydrogen, or whatever the calculated proportion may be. 
This refers to the composition of the interior not to any 
visible region and it is rather remarkable that there should 



20 40 60 

Percentage of Hydrogen 
Theoretical Constitution of the Sun 

be a way of discovering facts about the chemical constitution 
of such inaccessible matter. 

The method of determination is illustrated by the diagram. 
The curve represents the calculated brightness of the sun 
(strictly the heat-brightness) corresponding to different 
assumed percentages (by weight) of hydrogen. The full hori- 
zontal line represents the observed brightness. The crossing- 


points, where calculation agrees with observation, accordingly 
give the possible percentages of hydrogen in the sun. There 
are two crossing-points, one at 33 per cent, and the other at 
99-5 per cent. The first value seems more probable, especially 
when the corresponding results for other stars are taken into 
consideration; but there are some astronomers who advocate 
the other solution which exhibits the stars as globes of 
hydrogen with only a trace of the other elements. The 
second solution is ruled out if the heat of the stars is main- 
tained by the transmutation of hydrogen (p. 167), for the 
sun would have consumed more than 0-5 per cent, of the 
hydrogen during its past history. 

The diagram also shows two broken horizontal lines. 
These exhibit the uncertainty in the calculation which comes 
from our ignorance of how the subatomic source of the 
sun's heat is distributed through the interior. If it is all con- 
centrated at the centre, we must take the intersections with 
the upper broken line; if the subatomic energy is being 
liberated evenly all over the sun without regard to tem- 
perature, we must take the lower broken line. It is unlikely 
to approach either of these extremes, so that the uncertainty 
from this cause is not serious. 

A proportion of f hydrogen and \ not-hydrogen* seems 
to fit most of the stars that have been investigated. There is 
some indication that the very massive stars have still more 
hydrogen. The difficulty is to find enough stars with well- 
determined masses to test such questions satisfactorily. One 
thing seems clear: stars of the same mass contain a remarkably 
constant proportion of hydrogen. It is hard to understand 
this constancy. 

We cannot have it both ways; and since we use the com- 

* Owing to the lightness of hydrogen the proportion by weight 
scarcely gives a fair idea of its abundance. The proportion, | hydrogen 
and | iron, would mean that there are 28 hydrogen atoms for I iron 


parison of the observed and calculated values of the luminosity 
to determine the (otherwise unknown) proportion of hy- 
drogen, we cannot claim that the agreement furnishes any 
exact confirmation of the theory. Nevertheless there is a 
valuable check. It was by no means certain beforehand that 
any proportion of hydrogen would satisfy the observations. 
It will be seen from the diagram that if we do not know the 
proportion of hydrogen we can calculate a minimum bright- 
ness of the sun, corresponding to the lowest point of the 
curve. It is certainly a check that the observed brightness 
turns out to be above and not below the minimum brightness 
predicted by theory, and it is not so much above the mini- 
mum as to render the test a trivial one. The same test is 
satisfied by other stars covering a wide range of brightness, 
mass and density. When it is remembered that the minimum 
luminosity is calculated for the stars without knowing their 
chemical composition, without knowing how their heat is 
maintained, with no data except the mass and a rough know- 
ledge of the radius, and using only the properties of matter 
determined under totally different conditions in the labora- 
tory, it is very satisfactory to find that the actual luminosities 
of the stars run regularly a magnitude or two above the 
minimum calculated for them. 

Fate has been rather unkind. After ten years doubt we 
seemed to have settled satisfactorily with the troublesome 
element hydrogen. And just then the physicists must needs 
discover a new element, neutron, of an equally upsetting 
kind. In 1924 I had to make the reservation "provided that 
the stars do not contain an excessive proportion of hydrogen ". 
It seems that in 1934 1 must make the reservation "provided 
that the stars do not contain a significant amount of neutron ". 
The trouble is that neutron would make the material a very 
good conductor of heat. With any other material the leakage 
of heat through the star by conduction is negligible compared 
with the leakage by radiation. But it is said that 5 per cent. 


of neutron would so greatly increase the conductivity that 
the whole of the observed outflow of heat would be attri- 
butable to conduction. I doubt whether we yet know enough 
about neutron to justify this estimate; but we must certainly 
keep an eye on the newcomer. 

All the time our difficulty has been to construct a star 
which shall sufficiently hold in its internal heat, i.e. shall not 
be too bright. We have found it necessary to make it largely 
of hydrogen in order to keep the temperature low. I have 
busied myself with damming back the aetherial heat, but my 
efforts are of no avail if meanwhile neutron sneaks in and lets 
out the material heat. 

It seems rather probable that there is a saving circumstance. 
The neutrons, or atoms of neutron, easily enter atomic nuclei ; 
and presumably any neutrons evolved in a star will have only 
a brief free existence. We should expect them to be quickly 
snapped up by the atomic nuclei present, this being one of 
the processes of transmutation of die elements. Thus it may 
be hoped that the star will be kept clear of free neutrons, and 
that this threat to our conclusions will be countered. 


Referring once more to the title of Lane's paper, we notice 
the phrase "under the hypothesis of a gaseous mass". Lane 
knew quite well that the mean density of the sun is 1-4 times 
that of water; so that it was rather an astonishing proposition 
to treat the material as though it were a gas. The practical 
astronomer could scarcely be blamed if he paid scant attention 
to a theory which took such liberties with the plain facts of 
the problem. Long after Lane's time it was discovered by 
Russell and Hertzsprung that there is a class of stars, called 
"giant stars", to which the theory can safely be applied 
because they have low densities like an ordinary gas. Capella, 
for example, has a mean density about equal to that of the air 


around us. Betelgeuse and Antares are more extreme ex- 
amples of rarefied stars. By terrestrial standards we should 
describe Betelgeuse as "a moderately good vacuum". If our 
sun were distended to the dimensions of Betelgeuse it would 
envelop the earth's orbit, and we should be inside it. 

It came to be accepted that Lane's theory applied only to 
the giant stars; and the other great division, the "dwarf 
stars" whose densities are comparable with those of terrestrial 
solids and liquids, was deemed to be outside its scope. 
Similarly the more extended investigations that I have been 
describing, including the theoretical calculation of the lumi- 
nosity, assume that the star consists of perfect gas, and 
therefore the results were expected to apply only to giant 
stars. For such stars the luminosity depends on the mass, 
radius, and proportion of hydrogen; but the radius is com- 
paratively unimportant, since any change of radius within 
reasonable limits is found to make very little difference to 
the result. Since the consideration of the hydrogen content 
is a later refinement, the calculation presented itself primarily 
as a theoretical relation between the mass and the luminosity 
of a star. 

The mass-luminosity relation was calculated in 1924 and 
found to be satisfactorily obeyed by the giant stars. But the 
agreement went too far. Practically every star agreed with the 
formula the rarefied stars for which it was intended and the 
dense stars for which it was not intended. 

Consider, for example, the sun. If we make the rather 
strange assumption that its material is compressible like a 
perfect gas, its luminosity should be that given by the formula. 
But if its material is less compressible, it will hold its own 
against the pressure without requiring so much heat to keep 
it distended. The internal temperature will therefore be less 
than supposed, and not so much heat will leak out. Incom- 
pressible stars should therefore have a luminosity less than 
that given by the mass-luminosity relation. Conversely, if 


we find that a star obeys the mass-luminosity relation, that 
is evidence that its material is compressible like a perfect gas. 

The sun and other dense stars obey the formula calculated 
for a perfect gas. The plain conclusion is that they are com- 
posed of perfect gas. Something in the condition of stellar 
matter the extreme temperature or pressure must have 
made it possible for material as dense as water or as iron to 
yield to pressure in the same way that an ordinary gas yields. 

The explanation was not difficult to find. Why is it that 
we can compress air but we cannot appreciably compress 
water? It is because in air the ultimate particles the mole- 
cules are wide apart with lots of empty space between them. 
So when we compress air we are merely squeezing out this 
emptiness. But in water the molecules are practically in 
contact and there is no emptiness to be squeezed out. In all 
compressible substances the limit of compression is when the 
molecules begin to jam together. If a gas is compressed more 
and more, there comes a time when nearly all the empty 
space has been eliminated and the particles begin to jam; it 
then loses its characteristic compressibility and is sai4 to have 
become "imperfect". By that time its density is more or 
less that of an ordinary solid and liquid. This refers to our 
terrestrial experience. But in a star the bulky terrestrial 
atoms and molecules no longer exist; most of the satellite 
electrons are torn away by ionisation. The lighter atoms are 
reduced to a bare nucleus of almost infinitesimal size; the 
heavier atoms retain a few of the closest electrons forming 
a structure perhaps ~ of the diameter of a complete atom. 
So at the density of water or of the sun, when the complete 
atoms (if they existed) would be jammed in contact, there 
is still plenty of room between these tiny structures ; and 
jamming will not occur until the matter is compressed to a 
density of the order 100,000 times greater. The material of 
the sun is therefore very far from its limit of compressibility, 
and it is right that we should find it behaving as a perfect gas. 


Before this explanation could be accepted it was necessary 
to examine the effect of the electric charges of the ionised 
atoms. The effect is found to be small and actually tends to 
make the material slightly more compressible than a perfect 
gas.* This sounds paradoxical, for you would think that it 
would be more difficult to squeeze together ions which repel 
one another than complete atoms which are electrically 
neutral; but it must be remembered that the electrons which 
have been torn from their orbits are still present in the 
material, and they are able to compensate the nuclear charges 
as effectively as if they were bound. 

We are therefore led to the conclusion that in stellar con- 
ditions the limit of compressibility will not be reached until 
the matter is perhaps 10,000 times denser than anything 
known on the earth. The most effective confirmation of the 
theory would be to find such dense matter actually existing. 
It happened that we knew where to look for it. In certain 
stars three examples were known at the time the usual 
method of determining the density appeared to fail for some 
unexplained reason. For the Companion of Sirius the method 
gave the absurd density of 60 kilograms to the cubic centi- 
metre, or about a ton to the cubic inch. But if our theory is 
right, such a density is not necessarily absurd, and the method 
may not have failed after all. Accordingly W. S. Adams at 
the Mount Wilson Observatory undertook to check the 
deduced dimensions of the star, employing a method based 

* This is partly a question of definition of "perfection". When an 
ionised gas is compressed at constant temperature, the ionisation di- 
minishes. Neglecting die electrical forces, the reduced ionisation makes 
the pressure somewhat less than it would have been (at the same density) 
if the constitution of the gas had remained unchanged. Therefore, in 
comparison with an ideal gas of fixed constitution, we find that an 
ionised gas should be rather more compressible; the same increase of 
density corresponds to a smaller increase of jpressure. The electrical forces 
diminish this super-compressibility, but they still leave the material 
slightly more compressible than an ideal gas of fixed constitution. 


on Einstein's theory of gravitation. His results, which verifie< 
the high density, have since been confirmed at the Lie] 
Observatory. It is generally accepted that the Companioi 
of Sirius is an example of a star in which the material has ai 
average density 2000 times greater than platinum. A mate] 
box filled with this material would require a derrick to lift it 
since it would weigh about a ton. The dense material lies we] 
below the surface at a depth where there is sufficient super 
incumbent matter to supply die necessary pressure. There i 
nothing abnormal about the layers which we actually see. 
These super-dense stars are known as white dwarfs. The^ 
are probably very abundant in space; but since they hav< 
low luminosity we can only discover those that are in ou 
immediate neighbourhood. Only three are definitely re 
cognised, but several more stars have been assigned to thi 
class on more or less probable evidence. There is also fairb 
strong ground for believing that the nuclei of planetar 
nebulae are white dwarfs. In these stars the matter is to< 
near the limit of compression to be treated as a perfect gas 
and they do not follow the mass-luminosity relation. A 
they form an exceptional class of great rarity from the poin 
of view of the practical astronomer, we have not ordinarth 
the white dwarf condition in mind when speaking about th< 
stars in general; and it must be understood that statements ii 
which I attempt to convey the leading features of Stella 
constitution will not always apply to the white dwarfs. 


It happened that just about the time that super-dense mattei 
was discovered in the stars, an important development o 
wave mechanics was turning the thoughts of theoretica 
physicists in the same direction. R. H. Fowler was the firs 
to recognise that the white dwarf stars provided a field o 


application for the "new statistics" which, according to 
wave mechanics, replaces the classical statistics of the ordinary 
theory of gases when the particles become crowded together. 
His treatment of the dense matter in white dwarf stars has 
been developed and extended by many subsequent writers. 
The theory depends primarily on the famous law in quantum 
theory called Pauli's Exclusion Principle. In its more special 
form of application it asserts that two electrons in an atom 
cannot occupy the same orbit (p. 35). More generally it 
requires that there shall always be a certain minimum of 
distinction between one electron and another. If the dis- 
tinction had reference to position only, we could divide 
space up into equal unit cells and express the principle by 
saying that two electrons cannot be in the same cell. But the 
distinction also takes account of energy and momentum. 
That provides, as it were, another dimension in which dis- 
tinction is possible if the distinction in position is insufficient. 
For a rough picture we can imagine the positional cells to 
form the ground floor of a sky-scraper. Two electrons cannot 
occupy the same ground cell; but one of them can occupy 
the corresponding cell on an upper floor if it has the energy 
corresponding to that elevation. 

We imagine then the electrons to be living in a sky-scraper 
whose ground plan corresponds to space. The building is 
divided into rooms of uniform size, and a County Council 
regulation against overcrowding provides that two electrons 
may not occupy the same room. The electrons are moving 
and therefore continually changing their rooms. To mount 
to a higher floor the electron requires additional energy 
which it must absorb from radiation present in the material.* 
If it descends to a lower floor it emits energy. In a cold body 
at the absolute zero of temperature there is no radiation 
present; consequently the electrons may come down but they 

* Disregarding mere exchanges of energy between the electrons by 
which one mounts up at the expense of another going down. 


cannot mount up. In course of time they will all come down 
to the ground floor, provided there are enough rooms there. 

The novel feature of very high density is that there may 
not be enough rooms on the ground floor, so that some of 
the electrons have to remain in the upper stories for lack of 
room below. In the more familiar low-density conditions 
there may be electrons on the upper floors; but this is mere 
exuberance of spirits the result of a plentiful supply of 
energy. When the congestion on the ground floor begins, the 
pressure needed to compress the material is greatly increased, 
because it has not merely to pack the particles tighter but to 
lift them up to a floor where there is room for them. 

A peculiarity of matter in this congested or, as it is generally 
called, degenerate condition is that, although it contains a great 
deal of what we should naturally call heat-energy, it is quite 
cold. The electrons relegated to the upper stories have energy ; 
we picture them as travelling with great speed. But, unlike 
electrons similarly energised in uncongested conditions, they 
cannot spend their energy; it has, as it were, to be kept on 
deposit. Degenerate matter has thus a large latent heat, 
which is not available for radiation and does not take part in 
temperature exchanges. The latent energy can only be made 
available by allowing the matter to expand and become non- 

We cannot cover all the ramifications of the theory by an 
artificial picture of this kind: but the conception of a de- 
generate state of matter in which all the lower energy levels 
are filled up, and any additional particles forced in by com- 
pression have to be endowed with sufficient energy to occupy 
a high energy level, enters largely into the mathematical 
theory of dense stars. 

The question arises, Is a high temperature necessary for 
attaining the white dwarf condition of matter? Supposing 
that we could apply sufficient pressure, would it be possible 
to crush cold terrestrial matter to a thousand times the 


density of platinum, or would it be necessary first to 
smash the atoms thoroughly by heating it up ;o 10,000,000 ? 
It is now clear that pressure alone would suffice. The fragile 
shell of satellite electrons, which can be broken by the attacks 
of X rays or the fierce collisions in the interior of a star, can 
also break by simply giving way under the strain of pressure. 
Perhaps the strangest thing is that the compressibility of all 
kinds of matter whether its density be that of a gas, of a 
terrestrial solid, or of the Companion of Sirius is, apart 
from certain trivial aberrations, found to be much the same. 
There are two ways of reckoning the compressibility of 
material, according as the heat generated by the compression 
is or is not allowed to escape. We find the closest similarity 
if we adopt the second (adiabatic) reckoning. In a mon- 
atomic gas, e.g. helium, a 32-fold increase of pressure gives 
an 8-fold increase of density, if the heat of compression is 
retained in the gas. It is calculated that the dense matter in 
the Companion of Sirius is at least as compressible as this. 

Why do terrestrial solids and liquids stand aside from the 
general rule that matter has a compressibility of the same 
order as that exhibited by helium? That is the trivial aber- 
ration referred to above. In the long run dense matter is not 
less compressible than rarefied matter, only its compression 
proceeds more jerkily. The apparent incompressibility of 
terrestrial solids and liquids is due to the fact that the 
ridiculously small pressures available to man are insufficient 
to get over the first jerk. 


These people are under continual disquietudes, never enjoying a minute's 
peace of mind; and their disturbances proceed from causes which very 
little affect the rest of mortals. Their apprehensions arise from several 
changes they dread in the celestial bodies. For instance. . .that the sun, 
daily spending its rays without any nutriment to supply them, will at 
last be wholly consumed and annihilated; which must be attended with 
the destruction of this earth, and of all the planets that receive their light 
from it. SWIFT, Gulliver's Travels: A Voyage to Laputa. 

ARTIFICIAL transmutation of the elements was first accom- 
plished by Cockcroft and Walton in 1932*. Up to that rime 
our knowledge of the conditions of release of subatomic 
energy was derived almost wholly from astrophysical 
researches. In due time the data now being found in the 
laboratory will be of the utmost value to astronomy; we 
are on the threshold of big developments in the theory of 
stellar evolution and other problems depending on a know- 
ledge of the source of a star's heat. But in this discussion 
I do not want to give too much prominence to our first hasty 
reflections on the new situation. We must wait until the 
present riot of experiment wears itself out a little. I would 
rather show the progress that astronomy has been able to 
make with the problem by its own resources, reserving until 
the end of the chapter the question how far the results are 
supported by the new laboratory discoveries. It would be 
premature to claim that the astronomical conclusions have 

* Transmutation is produced by bombarding the nuclei with high- 
speed particles. By artificial transmutation we mean that the shower of 
bombarding particles is produced artificially. Some years earlier 
Rutherford had produced transmutations semi-artificially by using die 
high-speed particles emitted from radio-active substances. 


been definitely confirmed; but they appear to be in keeping 
with the present trend of physics, and the opposition which 
they long encountered has died down. 

Accordingly Sections i-m represent the outlook towards 
the end of 1932.* This enables us to introduce in Section rv 
the new experimental knowledge of the transformations of 
the atomic nucleus as entirely independent evidence bearing 
on the same questions. In so far as it is found (now or later) 
to lead to the same conclusions, it is a welcome corrobora- 
tion of the general ideas and methods used in the study of 
stellar constitution. 


I am going to tantalise you with a vision of vast supplies of 
energy surpassing the wildest desires of the engineer 
resources so illimitable that the idea of fuel economy might 
be put out of mind. We have not to travel far to find this 
land of El Dorado, this paradise of power; the energy to 
which I am referring exists abundantly in everything that 
we see and handle. Only it is so securely locked away that, 
for all the good it can do us, it might as well be in the 
remotest star unless we can find the key to the lock. We 
know very well that the cupboard is locked, but we are 
drawn irresistibly to peep through the keyhole like boys who 
know where the jam is kept. 

We build a great generating station of, say, 100,000 kilo- 
watts capacity, and surround it with wharves and sidings 
where load after load of fuel is brought to feed the monster. 
My vision is that some day these fuel arrangements will no 

* In order the better to recapture die ideas of the time, I have followed 
in these sections as closely as practicable the text of a lecture given to the 
World Power Conference at Berlin in 1930, omitting or condensing 
those parts which would duplicate explanations given elsewhere in this 
book, and introducing only the minor modifications necessary to bring 
the astronomical statements up to date. 



longer be needed; instead of pampering the appetite of the 
engine with delicacies like coal and oil, we shall induce it to 
work on a plain diet of subatomic energy. If that day ever 
arrives, the barges, the trucks, the cranes will disappear, and 
the year's supply of fuel for the power station will be carried 
in in a tea-cup, namely, 30 grams of water or 30 grams of 
anything else that is handy. 

I have called it a vision; but to the astronomer it means 
much more than an extravagant flight of theory. We look 
up at the sky and our telescopes show a thousand million 
stars. Everyone of these is a celestial furnace which apparently 
defies the law that limits our terrestrial undertakings that 
if you do not continually replenish your furnace it will die 
out. Geological, physical, biological evidence seems to make 
it certain that the sun has warmed the earth for more than 
a thousand million years; but the calculation first made by 
Kelvin still stands incontrovertible that the sun's heat cannot 
have been maintained for more than twenty million years 
unless it is being fed from some secret store of energy of a 
kind unknown in his day. By all ordinary rules the sidereal 
universe which we see blazing with light should have long 
since been cold and dead. None of the sources of power 
utilised by our present civilisation could have kept it alive 
for more than a small fraction of the time it is known to have 
existed. It seems then quite plain that the "cup of water" 
method of maintenance is actually in operation in the stars, 
or that there is some partial adaptation of it. To the engineer 
the prolific liberation of subatomic energy is a Utopian 
dream; to the physicist it is a pleasant speculation; but to the 
astronomer it is just a common well-recognised phenomenon 
which it is his business to investigate. 

As astronomers we have not merely to acloiowlecigc the 
existence of sources of subatomic energy; w? l^ve tojtudly 
observationally the laws of its release to cx^nu^K^r the 
fate of liberation of subatomic energy varies withjm.ejgjji-- 


perature, the density, or the age of the matter concerned. 
We "must also^ws^iST^w ..tb^.ajjj^ ,,keft v .UQ$!er 
cpntrot^C thitX^aFfecT witli heat in this way is not blown 
to pieces or thrown into violent oscillation. A few general 
fawrhave "been found in this way. It is true that they are 
only disconnected "fragments oF a complete scheme. But I 
have to insist that the study of subatomic energy is something 
imposed on us in the ordinary course of astronomical 
research, without which we cannot form any useful ^con- 
clusions as to the evolution and general functioning of the 
stars. Like many lines of research in course of active develop- 
ment it is still in an untidy and unsatisfactory state. 

Whilst insisting that it is a practical subject for the astro- 
nomer, I do not suggest that for the engineer it can be more 
than a dream for idle moments. I can see no escape from the 
conclusion that subatomic energy is the main fuel consumed 
in the celestial furnaces; but it would be wrong to raise 
illusive hopes that the astronomer may, like Prometheus, 
steal fire from heaven and make it available to men. Emerson's 
exhortation "Hitch your wagon to a star" is not to be 
followed literally by our transport authorities. 

I have referred to the practical utilisation of subatomic 
energy as an illusive hope which it would be wrong to 
encourage; but in the present state of the world it is rather 
a threat which it would be a grave responsibility to dis- 
parage altogether. It cannot be denied that for a society 
which has to create scarcity to save its members from 
starvation, to whom abundance spells disaster, and to whom 
unlimited energy means unlimited power for war and 
destruction, there is an ominous cloud in the distance though 
at present it be no bigger than a man*s hand. 

1 1-2 



Before jumping to the conclusion that the stars are utilising 
subatomic energy, there is a preliminary point to settle. I< 
it not possible that a star may be picking up from outside 
the energy necessary to maintain its radiation? Some have 
suggested that the sun is kept hot by meteors falling into it ; 
others that it collects cosmic rays or still more subtle forms 
of energy traversing space. In short the question is, Does the 
star live on extraneous power like a windmill, or does it 
contain its energy stored inside it like an accumulator battery r 
I think that all theories which postulate an outside source can 
be dismissed, because they misconceive the nature of the 
problem. It is the temperature of some millions of degrees in 
the central regions that has to be maintained, and this requires 
the generation of heat in the deep interior. Meteors anc 
cosmic rays provide only for keeping the surface hot. That 
is no use. For the maintenance of the sun's surface at 6ooo c 
would not stop the energy flowing out from the intensely 
hot interior, and the whole interior would presently cool 
down to the same temperature as the surface. We have seen 
that the internal heat is necessary in order to keep the sun 
distended to its observed volume. The problem of maintain- 
ing the sun's radiation is thus merged in the larger problem 
of maintaining its volume and other characteristics. We can 
only keep the interior of the star at a temperature of the order 
10,000,000 by providing a source of energy in the deep 
interior. It appears then that a star contains within it the 
fuel that has to last it for the whole of its life. 

The total store of energy contained inside the sun is easily 
calculated. Einstein has shown that there is an exact equiva- 
lence of mass and energy, such that i gram of mass represents 
9 . io 20 ergs of energy. (Tlie numBer 9 . io 20 is the* square 63 
the velocity of light in C.G.S. units.) We have only to convert 
the sun's mass 2. io33 gm. into ergs by this factor; the result 


i-8. io 54 ergs is the sun's whole stock of energy. We know 
by observation that the sun squanders 1-2. lo^ 1 ergs every 
year by radiating heat and light into space; thus the whole 
stock amounts to just 15 billion (1-5 . 10*3) years' supply. By 
passing from the ordinarily known sources of energy to this 
store of subatomic energy we extend the possible life of the 
sun about a million-fold. 

This does not mean that the sun will last 15 billion years 
and then go out; it is not quite so simple as that.. Xhe energy 
which we are now considering is energy of constitution .of 

, &_ ,^~ , . i . 

matter; and, or course, if you remove energy which is 
essential to its constitution the matter can no longer exist; 
it, so to speak, comes unstuck. And so as the stock of energy 
of the sun disappears little by little, the matter or mass of the 
sun disappears little by little. By the mass-luminosity relation 
(p. 153) the lower mass involves a lower rate of radiation. 
So we must allow for the fact that the sun will become less 
spendthrift in its old age; and its life as a waning star can be 
prolonged much beyond 15 billion years. 

Similarly we can show that the sun cannot be more than 
5 billion ( 12 ) years old. Large masses radiate very 
strongly; and however large a mass it started with, the sun 
would have radiated itself down to its present mass within 
5 billion years. I have never heard of any theory which 
required a longer past for the sun than that; but if anyone 
should propose a greater duration I think that astronomers 
would be justified in opposing it emphatically. It is more 
likely that we shall have to be content with a past duration 
of the sun very much less than this maximum estimate. 

We have here been assuming a very drastic process of 
liberating subatomic energy, involving the complete dis- 
appearance of matter into radiation. For the matter to 
disappear it must be supposed that the protons and electrons 
of which it is composed have the power of mutually de- 
stroying one another. The proton carries a unit positive 


charge and the electron a unit negative charge, and it may 
be that under certain circumstances two such opposite 
particles can coalesce and cancel out. The idea is that when a 
proton and electron run together and neutralise each other, 
nothing is left but a splash in the aether representing the 
energy of constitution which is now set free. The splash 
spreads out as an electromagnetic wave, which is scattered 
and absorbed until it is converted into the ordinary heat of 
a star. The process is not difficult to imagine, but it is open to 
doubt whether it actually occurs in Nature. Apart from an 
indirect, and now very unlikely, inference from the pheno- 
mena of cosmic rays, there has been not the slightest 
observational evidence of its occurrence. Nor can it be said 
that it is a theoretical necessity that it should occur. It is just 
a conjecture. On the other hand I am not sure that it is more 
speculative to suppose that protons and electrons can end 
their existence in this way than to adopt the contrary view 
which supposes them to be immortal.* 

There is a less drastic alternative. It is possible for matter 
to liberate some of the energy contained in it without going 
to the length of complete suicide. This alternative process is 
transmutation of the elements. By rearrangement of the 
protons and electrons in atomic nuclei a quite considerable 
amount of energy can be furnished. The most familiar 
example of such transmutation is in radio-activity. But none 
of the spontaneous radio-active transformations uranium 
into radium, radium into lead, etc. yields anything like 
enough energy for our purpose. Moreover it seems almost 
certain that a star is a place where radio-active elements are 
being synthesised, not where they break down. If the energy 
of radio-activity is of any account at all it must be reckoned 
a source of loss rather than of gain to the star; because 
presumably the transmutation is there proceeding the opposite 
way from that on the earth. 

* Sec, however, p. 181. 


The transmutation which might furnish sufficient energy 
to maintain the heat of the stars is the building up of complex 
elements out of hydrogen, more especially the formation of 
helium out of hydrogen. A hydrogen atom consists of one 
proton and one electron; a helium atom consists of four 
protons and four electrons, the four protons and two of the 
electrons being cemented together to form the helium 
nucleus. The material of a helium atom is thus precisely the 
material of four hydrogen atoms. But although the material 
is the same the mass is not quite the same; the helium is 
lighter by about i part in 140. By Einstein's law of the 
equivalence of mass and energy, this mass-defect is a measure 
of the energy that must be liberated when hydrogen is 
transmuted into helium. 

You see then that there are two conceivable ways of 
getting energy out of four hydrogen atoms. They may 
disappear totally, each electron cancelling a proton; in that 
case the whole mass is lost, and the whole energy of con- 
stitution is set free to maintain the stellar furnace. Or they 
may rearrange themselves to form a helium atom; in that 
case 4o of the mass is lost and 1^5 of the whole energy is set 
free. Slightly more energy is set free if hydrogen is trans- 
muted into a heavier element, e.g. oxygen, instead of into 
helium, but the advantage is trivial. Correspondingly the 
energy released in the transmutation of helium into oxygen 
is relatively insignificant. Recalling our previous classifica- 
tion of stellar matter as hydrogen and not-hydrogen, the 
only important source of energy is the transmutation of 
hydrogen into not-hydrogen; and this releases rather less 
than i per cent, of the whole energy. So if we decide to 
adopt the transmutation theory we must arrange to run the 
stellar furnaces on 7 of the fuel available according to the 
annihilation theory. This cuts down the time-scale in the 
ratio 755 , and brings down the maximum life of the sun from 
birth to death to 150,000 million years. I daresay we can 


make that suffice. It is a more generous allowance than we 
need according to the results of Chapter x. 

I have been speaking of the formation of helium and other 
elements out of hydrogen as though it were an established 
fact. It is true that no one has yet (1932) succeeded in per- 
forming such transmutations; but this objection seems 
scarcely relevant. It is an established fact that we find in 
Nature aggregations of protons and electrons in the par- 
ticular formation which is called helium; and we are only 
applying the ordinary scientific outlook when we regard 
such a formation as brought about by the operation of 
physical law and not by an act of special creation. The 
evolution of our ordinary atoms out of their constituent 
electric charges must have occurred at some rime and place. 
What place is more appropriate than the interior of a star, 
where the energy released by the process would serve for the 
maintenance of a star's heat? Can you suggest a more likely 
site for Nature's workshop where she forges a diversity of 
material out of the primitive basis of positive and negative 
electric particles ? I have often encountered critics who argue 
that the stars are not hot enough for this purpose. I once so 
far forgot myself as to tell the critic to go and find a hotter 

I will try to explain why it makes a big difference to 
astronomy which of the two possible sources of subatomic 
energy is in operation. Suppose first we assume the less 
drastic hypothesis of transmutation. Then i per cent, at the 
most of the total store of energy in the star is available to 
maintain its heat. Our peep through the keyhole showed us 
100 pots of jam on the shelves; but it has turned out that 
99 of them are unfit for consumption. As soon as it begins 
to shine, the star starts using up the one consumable pot, 
losing the corresponding amount of mass. When it has 
radiated away I per cent, of its original mass, the supply is 
finished; the furnace must die out and the star become cold. 


Thus the mass of the star remains constant to within about 
i per cent, during its whole history. Contrast this with the 
other hypothesis of annihilation of electrons and protons. 
All the mass is then consumable. The dying out of the furnace 
is postponed, and the star may live to radiate 50 per cent., 
75 per cent., 90 per cent, of the mass it had to start with. 
Beginning as a heavy-weight star it will gradually change 
into a light-weight star. By the mass-luminosity relation its 
brightness will diminish as its mass diminishes. We shall 
have an evolution of small stars from large stars, of faint stars 
from bright stars. Many interesting astronomical results 
arise out of tracing the consequences of this evolution. But 
all this falls to the ground if we reject the annihilation 
hypothesis and admit only transmutation. There is then no 
appreciable change of mass, and small stars differ from large 
stars because they were born different. So until we can 
decide between the two hypotheses we are like children 
speculating whether ponies grow into horses or whether 
ponies and horses have always been different. 

It suggests itself that we should try to make an observa- 
tional test whether big stars turn into little stars. It may not 
be an infallible test, but it is a fairly direct test. Let us take 
all the brand-new stars we can find, and see what sort of 
mass they have. I think it is fair to assume that the most 
recently formed stars are those with low density; we believe 
that the stars have condensed out of nebulous material, so 
the first stage should be a huge diffuse globe a star such 
as Betelgeuse or Antares. Taking a list of about 300 of the 
most diffuse stars and calculating their masses from their 
brightness by the mass-luminosity relation, we find that their 
mean mass is 3*6 times that of the sun, and nine-tenths of 
them are between si and 2\ times the sun's mass. It appears 
therefore that the stars at birth seldom or never have a mass 
so low as that of the sun. On the other hand taking stars of 
all ages there are far more masses below that of the sun than 


above it. It would seem that these must have lost a great 
part of their original mass, having radiated it away in the 
course of billions of years as the annihilation theory suggests. 

Unfortunately this is counterbalanced by evidence of 
another kind which is unfavourable. We sometimes find 
clusters of associated stars, which evidendy have a common 
origin and must have been formed about the same time. The 
Pleiades is a well-known example. The theory requires that 
these coeval stars should be of nearly the same mass and 
brightness. For if the cluster is young, there has not been 
time to radiate the large original masses down to the sun's 
mass or lower. If the cluster is old, the original range of mass 
will have been lessened in the course of time; because the 
large stars radiate away their mass very quickly and so tend 
to catch up the smaller stars which radiate more slowly. But 
this is not at all in accordance with observation. In the 
Pleiades the stars range over at least 10 magnitudes indicating 
a wide diversity of mass. We have to admit that, in the 
Pleiades at least, small stars are born small and not evolved 
out of big stars. Such exceptions make us very sceptical 
about the whole idea. 

One could cite other considerations of a similar kind, some 
rather favourable to, others rather against, the annihilation 
hypothesis. It is all very inconclusive. In studying the stars 
as individuals there is, in spite of some difficulties, much that 
attracts us to the hypothesis and the extremely long time- 
scale which results from it. But when we turn to consider 
systems of stars clusters and galaxies all the evidence 
indicates much less antiquity. It now seems very unlikely 
that we have to go back more than 10,000 million years. 
I am sorry to be so vacillating in one argument putting the 
beginning of things as we know them billions of years ago, 
and a few pages later lopping off two or three noughts from 
that figure; but everything depends on which line of circum- 
stantial evidence you trust. 


We shall see in Chapter x that the rapid expansion of the 
universe points strongly to the shorter time-scale. Additional 
support is given by a study of the dynamics of our own galaxy. 
It can be shown that the rotation and distribution of stellar 
motions which we find in our galaxy is incompatible with 
a strictly steady state of the system ; and it appears that change 
and dissipation must be fairly rapid.* Whereas a star con- 
sidered by itself is something which, so far as we can tell, 
might hkve existed with very little change for untold ages, 
the vaster systems clusters of associated stars, our own 
galaxy, and the whole super-system of the galaxies are in 
much more of a hurry to get on with their evolution. They 
are not yet worn down to regularity, and bear the marks of 
comparatively recent origin. Comparatively recent, I may 
remind you, means in this connection something of the order 
10,000 million years instead of the alternative suggestion of 
10,000,000 million years. 


Leaving the decision between the two possible sources of 
subatomic energy unsettled, we now consider the astro- 
nomical evidence as to the conditions which govern its 
release. On this side of the problem we seem to have quite 
definite information if only it were not so incredible! 
Apparently if you want to tap a really large supply of energy 
you must heat matter up to a temperature of about 20,000,000 
Centigrade. I will not guarantee that 20,000,000 is exactly 
the right figure; it may be 15,000,000 or a little less. But 
my point is that there is a temperature somewhere about 
this magnitude at which matter yields up its energy pro- 
lifically, whereas one or two million degrees below it the 
yield is practically nil. It is almost like a boiling point. 

* This is discussed in my Halley Lecture, The Rotation of the Galaxy 
(Oxford, 1930). 


The stars are now classified into three groups, the giants, 
the main series, and the white dwarfs. The giants are com- 
paratively few in number and presumably represent an early 
and rather transient phase of development. The white dwarfs 
are probably numerous, but owing to their low luminosity 
very few are actually known to us. By far the majority of 
the stars that we investigate belong to the main series. The 
main series forms a continuous sequence extending from the 
brightest to the faintest stars known. It is found that, from 
the top to the bottom of the series, the central temperature 
(calculated by the methods explained in Chapter vn) remains 
practically constant at about 20,000,000. At the top of the 
series we have very bright and massive stars radiating 10,000 
times as much energy as the sun; to keep up this output they 
require a continual supply of released subatomic energy 
amounting to 1000 ergs per second per gram of material. 
Near the middle of the series we have the sun, which requires 
2 ergs per gm. per sec. to maintain its output. At the bottom 
we have stars requiring -01 erg per gm. per sec. But whether 
the amount required is 1000 or 2 or *oi ergs per gm. per sec., 
the temperature has had to rise to 20,000,000 in order to 
set it free. 

So long as the subatomic energy liberated in the interior 
is less than the amount of energy squandered in radiation the 
star must go on contracting; and if it is an ordinary star (not 
a white dwarf), its internal temperature will rise. The rise 
continues until the conditions become such that the necessary 
amount of subatomic energy is liberated. When this balance 
is reached the star remains practically steady for an enor- 
mously long period, and we should expect to find the 
majority of the stars in this state. On the annihilation 
hypothesis the mass gradually diminishes and the star travels 
slowly down the main series. On the transmutation hypo- 
thesis the star remains stationary on the line of the main 
series until its hydrogen is mostly used up; presumably it 


then passes on to the white dwarf stage. In either case it is 
clear that a very rapid increase in the liberation of subatomic 
energy must set in at about 20,000,000, since stars requiring 
widely different amounts find their balance at about this 

Is this the key to the cupboard ? Suppose we could manage 
to heat terrestrial matter up to 20,000,000, should we extract 
its energy of constitution? I may remark in passing that if 
this is the method required, the chances of our making a 
commercial success of it are not very promising; we should 
waste a lot of power in maintaining the high temperature 
whilst the stream of subatomic energy dribbles out. But 
I scarcely think it can be the key. It must have some bearing 
on the problem; but a more general survey of the difficulties 
than I can give here convinces me that there is a great deal 
more that we shall have to understand before we can put 
these astronomical results in their right perspective. Twenty 
million degrees is perhaps not beyond attainment in our 
laboratories. At the Cavendish Laboratory Prof. Kapitza 
produces momentary magnetic fields in which the concen- 
tration of energy corresponds to about 1,000,000. If he 
should be able to raise this to 20,000,000 Well, I have said 
that I do not really expect the subatomic energy to come 
pouring out; but all the same I shall not go too near the 
laboratory when the experiment is tried.* 

There is another condition of release which is of great 
importance in astronomy. It is necessary that the liberation 

* This was written when we had no theoretical knowledge as to the 
cause of the critical temperature, and the possibility that it might be a 
genuine "boiling point" had to be reckoned with. I think there is no 
doubt that matter containing hydrogen, e.g. water, is a high explosive 
in the sense that the sudden generation of sufficiently high temperature 
would release subatomic energy so fast that the temperature would be 
maintained and spread through surrounding matter regeneratively; but 
it now seems clear that the regenerative temperature is considerably 
higher than 20,000,000. (See Section iv.) 


of subatomic energy should be stimulated by increase of 
temperature; otherwise it will not automatically adjust itself 
to keep the star steady for long periods of time, and subatomic 
energy will therefore fail to serve the purpose for which we 
have introduced it. But it must not increase too fast with the 
temperature, because that would have the effect of throwing 
the star into pulsation. It is very probable that some stars 
do pulsate, alternately swelling out and contracting in a 
period of a few days or hours; they form a class of variable 
stars called Cepheid Variables. At one phase of the pulsation 
the star's material is compressed and hotter than the average; 
at the opposite phase it is expanded and cooler. The subatomic 
supply of heat will be stimulated by the increased tempera- 
ture at compression and reduced by the lower temperature 
at expansion. Now this is just the way in which the heat 
supply of an engine must be regulated in order that the engine 
may be set working; heat must be supplied to the cylinder 
at compression and removed at expansion. Thus the star 
becomes an automatic engine which can maintain its own 
pulsations, or even work up a large pulsation out of a very 
small initial disturbance. The puzzle is, not to explain the 
Cepheid Variables, but to explain why they are the exception 
and not the rule. 

The pulsation will be attended by a certain amount of 
wastage; and the occurrence of the pulsation depends on 
whether the engine-effect that I have described is strong 
enough to make good the wastage. If the resistance is too 
great an engine will not start up. We must suppose that in 
the sun and in ordinary stars the engine is not strong enough 
to keep a pulsation going. That is one of the conditions to 
which we have to attend in formulating the laws of release 
of subatomic energy; they must not provide too powerful 
an engine. In other words the release must not be stimulated 
too rapidly by a rise of temperature above the normal 
temperature in the star. We can calculate roughly how rapid 
a rate of increase with temperature is permissible. 


This puts us in a dilemma. By comparing the temperatures 
of the various stars on the main series, we have seen that the 
increase in the rate of liberation of subatomic energy from 
01 to 1000 ergs per gm. per sec. must occur within a range 
of temperature too small for us to detect with certainty, say 
two or three million degrees. This is much too rapid a rate 
of increase to satisfy the new condition that we have found. 

Apparently the only way out of this difficulty is to suppose 
that the stimulating effect of an increase of temperature is 
delayed. There must be a time lag anything from a few days 
to a thousand years between the rise in temperature and the 
corresponding increased output of energy. That is to say, 
when the increase of temperature occurs there is no great 
immediate increase in the production of energy, but there is 
an increase in the production of an active kind of material 
which in due course (after some days or perhaps years) 
undergoes a spontaneous transformation which liberates 
subatomic energy. Such a time lag would smooth out the 
effect of the rapid changes of temperature in a pulsation; for 
it makes the rate of liberation of energy depend on the average 
temperature during the period of the lag. This will save the 
star from being thrown into pulsation. On the other hand 
it would make no difference to the permanent adjustment of 
the rate of liberation of subatomic energy to the rate of 
radiation of the star. 

We can now sum up the astronomical evidence concerning 
the liberation of subatomic energy: 

(1) There is abundant liberation of some form of sub- 
atomic energy at a comparatively low temperature of the 
order 20,000,000 or rather less. 

(2) Unlike ordinary radio-activity it is affected by the 
physical conditions of the material, and the liberation in- 
creases very rapidly with the temperature. 

(3 ) There is a time lag between the change of temperature 
and the corresponding change in the rate of liberation. This 
signifies that unstable material is formed which after some 


days or years spontaneously breaks down; and it is in this 
subsequent break-down that the greater part of the energy 
is liberated. 

(4) Evidence as to whether the source of the energy is 
transmutation of hydrogen or annihilation of protons and 
electrons is inconclusive; but the recent tendency is to favour 
the former with its accompanying short time-scale of 


Among physicists generally there was a great reluctance to 
accept the conclusions (i) and (2) in the foregoing summary. 
There were also astronomical critics. It was continually 
urged that subatomic processes could only be influenced by 
temperatures a thousand or a million times greater than those 
which we have found in the stellar interior. It is for this 
reason that I have called 20,000,000 a "comparatively low^ 
temperature". Throughout the last fifteen years there have 
been attempts to find a loophole for attributing a much 
higher temperature to the centre of a star, or alternatively 
to work the machinery of a star with an unadjustable (radio- 
active) source of energy unaffected by temperature and 
density. These have seemed to me to ignore one or more of 
the essential conditions of the problem, and to subordinate 
that branch of the subject the mechanical and thermal 
equilibrium of the star which depends on fairly well-known 
laws of physics to speculation on matters about which the 
physicist knew even less than the astronomer. 

The recent achievement of artificial transmutation of the 
elements in the laboratory has brought about a revulsion of 
feeling, and the astronomically determined temperatures of 
the stars are no longer criticised as too low. The transmuta- 
tion is accomplished by subjecting the atomic nuclei to 
bombardment by particles of various kinds protons, 
electrons, neutrons, doitons, helium nuclei (a particles). 


A certain proportion of these hit the nuclei and enter them. 
The particle may simply be retained, or its ingress may upset 
the equilibrium of the nucleus in such a way that some other 
kind of particle is expelled; in either case the constitution of 
the nucleus is changed and it becomes a different element. 
Here we are chiefly interested in the entry of protons 
(hydrogen nuclei) into the more complex nuclei, for we 
have seen that the astronomically significant liberation of 
energy (if any) comes from the transmutation of hydrogen. 
It is found that no great energy is required to enable a proton 
to penetrate a nucleus; the lowness of the energy seems to 
have come as a surprise to the experimenters. The progress 
of artificial transmutation in 1933 was made possible not by 
the use of unprecedentedly high voltages but by the great 
advance in die sensitivity of the methods of detecting 

The average energy of the particfes (including the protons) 
A jx,ar the centre of the sun is equal to that imparted to a proton 
when about 2500 volts are applied. There will always be a 
few protons with energies many times greater than the 
average comparable therefore with the energy of the 
protons employed in artificial transmutation. We do not 
want the conditions to be such that the protons enter the 
nuclei very often, for the sun's supply of hydrogen has to 
last it for at least io 10 years. Until more detailed laboratory 
data are available it is impossible to make a precise com- 
parison, but the general estimate is that at somewhere be- 
tween 10,000,000 and 20,000,000 the protons (or hydrogen) 
would disappear into the nuclei quite fast enough to provide 
the energy used to maintain the sun's heat. The transmutation 
is very sensitive to an increase of voltage, and correspondingly 
to an increase of temperature; so that stars requiring widely 
different supplies of energy will find their equilibrium at 
temperatures within a rather small range. Thus the observa- 
tional result which at first seemed so incredible is confirmed. 
ENPS 12 


It should be noted that even if we prefer the hypothesis of 
annihilation of electrons and protons as the main source of 
a star's energy, we must not disregard the effects of trans- 
mutation of hydrogen. The transmutation of hydrogen will 
act as a buffer preventing the temperature from rising above 
20,000,000 so long as any appreciable amount of hydrogen 
remains in the star. For if the star contracts so as to raise its 
temperature, the protons will attack the atomic nuclei more 
frequently; more energy will be liberated, which will cause 
the star to expand again and the temperature to fall. Theories, 
advocated until recently, which attributed temperatures of 
thousands of millions or billions of degrees to the stars, are 
now quite out of the question unless the stars arc assumed 
to be almost devoid of hydrogen in their interior. At such 
temperatures matter containing hydrogen would be a high 

There is as yet no direct confirmation of the time lag in 
the liberation of the energy (p. 175). On the other hand it 
is no longer a surprising conclusion; for in the bombardment 
of atomic nuclei with various particles (but, I think, not as 
yet with protons) it is often found that unstable nuclei are 
created which break down and give out energy after a few 
minutes or hours. 

The new discoveries may perhaps have removed one of 
the difficulties in the conception of evolution of complex 
elements inside a star. Formerly we knew of nothing inter- 
mediate between a proton and a helium nucleus. Thus the 
first step in evolution appeared to be the gathering together 
of 4 protons and 2 electrons to form a helium nucleus. How 
these could assemble simultaneously at one spot baffled 
imagination. We could only comfort ourselves with the 
reflection that they obviously had managed to assemble, and 
that the interior of a star could scarcely be a less favourable 
place for the purpose than anywhere else. But now neutrons, 
deutons, and isotopes both of hydrogen and of helium of 


weight 3, have been discovered, all intermediate between a 
proton and a helium nucleus. Thus helium may be built up 
gradually by the same kind of steps that occur in the evolu- 
tion of the higher elements. 

An alternative possibility (suggested and developed by 
R. D'E. Atkinson in 1931) is that the helium is formed inside 
complex nuclei and then expelled. To take an ideally simple 
example, we can suppose that protons and electrons enter a 
complex nucleus one by one, where they arrange themselves 
as far as possible as a particles. Now and then the structure 
collapses and an a particle (helium nucleus) is expelled. This 
may happen over and over again in the same nucleus. 
Assuming that, if the helium nucleus is accounted for, there 
is no difficulty in its further transmutation into a more 
complex nucleus, the progeny of helium nuclei will in due 
time provide additional complex nuclei to carry on the work. 
A single helium atom might in this way be the ancestor of 
all the not-hydrogen in a star. 

At one rime it seemed that Cosmic Rays might have an 
important bearing on the problem of subatomic energy. 
Cosmic rays is the name given to a highly penetrating 
radiation (consisting either of electromagnetic waves or 
particles) which travels downwards through our atmosphere, 
apparently having come into it from outside. It has been a 
favourite hypothesis that they have their birth in subatomic 
processes occurring in the nebulae or cosmic clouds in our 
own and other galaxies; they have been variously attributed 
to the transmutation of hydrogen into particular elements or 
to the annihilation of electrons and protons. Attempts to 
identify the process originating them depend on our knowing 
the energy of an individual ray, and until recently this could 
only be inferred from measurements of the penetrating 
power. It now appears that the energy of the strongest rays 
was very much underestimated, and previous interpretations 
have had to be revised. When stopped by matter a cosmic 



ray sometimes produces a great shower of electrons and 
positrons*; these can be traced individually in a Wilson 
expansion chamber and their energies of projection measured 
and summed. The original energy of the ray must be not less 
than this total. It turns out to be very much greater than the 
energy of any individual subatomic process admitted by 
existing theory. The cosmic rays are still a great mystery; but 
in view of their excessive energy it now seems impossible to 
attribute to them a subatomic origin. 

In dismissing cosmic rays from our subject we must 
dismiss with them certain ideas for which they were re- 
sponsible. It was clear that they could not come from the 
hot interior of a star, because they could not pass through 
any considerable part of the thickness of the star. They had 
therefore to be attributed to diffuse matter through which 
they would have practically free passage. The observed 
intensity of the cosmic rays indicated that the comparatively 
cool diffuse matter of the universe must be liberating energy 
not much less abundantly than the stars themselves. Against 
the natural conclusion from stellar observation and theory 
that the liberation of subatomic energy depends on the rather 
high temperature in the interior of the stars, we had to set 
the apparent evidence of the cosmic rays that high tempera- 
ture is by no means essential inasmuch as similar liberation 
occurs in nebulae. The latter evidence has proved untrust- 
worthy, and there is now nothing to distract us from the 
stellar clues. 

The discovery of the positron deals a blow to the annihilation 
hypothesis. We now know that the positron, not the proton, 
is the true enantiomorph of the electron. A positron and an 
electron can annihilate one another. The experimental evidence 
seems conclusive that twin electrons and positrons are created 

* See Plate I. 



when radiation of sufficient energy falls on matter, and that 
after a brief existence the positron ends its life by mutual 
suicide with an electron. Of course, this does not prove that 
an electron cannot equally end its existence by cancelling a 
proton; but the hypothesis begins to look rather gratuitous. 

The discovery of the neutron also makes a difference. One 
has the feeling that the combination of proton and election 
in a neutron is the nearest they can go to cancelling one 
another. In a sense it is not far off cancellation, for the neutron 
is, as we have seen, "an isotope of nothing". A neutron is so 
elusive, and has so little interaction with the matter through 
which it passes, that it is hard to detect that there is anything 
there. Having discovered this form of intimate combination 
of a proton and an electron a state of zero quantum number 
we feel it unlikely that there is yet another kind of com- 
bination resulting in complete destruction. 

To this I may perhaps add a personal view, based on the 
way in which the combined relativity and quantum theory 
is working out, that there are conditions which fix for all 
time the net number* of electrons and protons in the 

Although I have not ventured to go so far elsewhere in 
this book, I think the time has come to consider whether the 
hypothesis of annihilation of electrons and protons might not 
be allowed to lapse. I can perhaps suggest this the more 
freely because I think that as an astronomical hypothesis it 
first occurs in my own writings, f although the general idea 
was familiar enough to physicists at the time. As in the case 
of determinism, it is not a question of asserting definite 
disproof, but of realising that it is no more than a survival 
from a time when the state of our knowledge was different 
from that prevailing to-day. When the hypothesis was first 

* Counting a positron as "minus an electron", and a negatron as 
"minus a proton* . 
f Monthly Notices of the R.AS., vol. 77, p. 611 (1917). 


suggested no other adequate means of maintaining a star's 
energy was known. It was not until 1920 that Aston's 
accurate determination of the atomic weight of hydrogen 
revealed the large amount of energy to be obtained by the 
transmutation of hydrogen into not-hydrogen, and so pro- 
vided a possible alternative. We have seen (p. 168) that a 
decision between the two alternatives was not to be under- 
taken lightly, owing to its profound effect on our views of 
stellar evolution; and indeed the annihilation hypothesis was 
at the time the more conservative, being less disturbing to 
the current theory. Since then the relative status of the two 
hypotheses has changed in the following ways : 

(1) Transmutation is now a matter of practical know- 
ledge and is studied in detail in the laboratory. It is known 
to occur in conditions corresponding to the temperature of 
the stars. We have in any case to take account of its effect on 
a star's supply of energy, whether or not it is the sole source. 
On the other hand there is no observational evidence of 
annihilation; the cosmic rays which were sometimes du- 
biously regarded as giving such evidence are now found to 
have a different origin. 

(2) It now appears inevitable that we should accommodate 
ourselves to the shorter time-scale, and the main advantage 
of the annihilation hypothesis disappears. Accepting 10,000 
million years as an upper limit to the age of the stars, the 
sun's heat would be maintained for this period by trans- 
muting an amount of hydrogen equal to 10 per cent, of its 
mass. In this connection the discovery of the great abundance 
of hydrogen in the stars (p. 147) is a favourable point. 

(3) From the theoretical point of view the cancelling of 
an electron and proton is not so natural a suggestion as it 
formerly appeared. Larmor's picture* of the creation of a 
positive and a negative particle by rotating the walls of a 
tube with respect to an inner core with the possibility that 

* Aether and Matter, Appendix E (1900). 


the walls may ultimately slip back, annihilating the two 
particles is now seen to refer to the electron and positron 
rather than to the electron and proton. 

The present moment, when there is a rush of new discovery 
only half digested, is not the best time for making up our 
minds whether the hypothesis of annihilation is worth 
preserving. It will be apparent from many passages in this 
book that I have not yet taken the step of retiring it from 
my own thoughts. It is doubtless best to leave the question 
in abeyance for a year or two longer, but it has seemed well 
to call attention to its imminence. 


When I behold, upon the night's starr'd face, 
Huge cloudy symbols of a high romance. 

KEATS, Sonnet. 

I AM going to speak about a very rarefied cloud of gas which 
occupies all the space between the stars. First let me remind 
you of the vastness of this space and the extreme isolation 
of the stars from one another. The stars are small oases of 
matter in a desert of emptiness. For a traveller in this desert 
we may take a ray of light. His journey from one oasis to 
the next, say, from the nearest star to our sun, takes four 
years; he takes only eleven hoursNto cross the whole extent of 
the solar system; and then the journey is through empty 
desert again for six years or so. That is if the light ray were 
to zigzag from star to star; if it goes unheedingly on a straight 
course through the universe it will probably miss the oases 
altogether as a traveller in a desert would do. 

But this space between the stars, which I have called a 
desert of emptiness, is not entirely empty. There are vestiges 
of matter everywhere. In some parts of the heavens we can 
actually see a rarefied cloud amidst the stars. Examples are 
shown in Plates 2 and 3. In one the nebulous matter is bright 
wisps of glowing gas wreathed into a delicate lacework. 
hi the other there is, besides bright matter, an impenetrable 
black cloud blotting out everything behind it. It is only in 
:ertain regions that we see it thus plainly, but the cosmic 
matter extends everywhere. The recognisable nebulae are 
Condensations places where the density is sometimes as 


Mount Wilson Observatory 


much as a thousand or ten thousand times the normal. I shall 
first speak of the normal regions, where accordingly the 
photographs give no indication of matter being present. The 
invisible gas filling these regions will be called the "cosmic 
cloud" or "interstellar cloud". We ourselves are probably 
in a normal region where the cloud has more or less its 
average density. 

Until about ten years ago astronomers had no very satis- 
factory evidence of the existence of the cosmic cloud; never- 
theless it has been a subject of discussion for forty years or 
more. Our former attitude towards it reminds me of the 
guest who objected to sleeping in the haunted room. "But 
I thought you did not believe in ghosts!" "I don't believe 
in ghosts, but I am afraid of them." Probably not many 
astronomers believed in the cosmic cloud, but some of them 
were afraid of it. Afraid, because, if such a feature of the stellar 
universe existed unheeded in our calculations, it might upset 
some of our most fundamental conclusions in astronomy. 
Having measured the apparent magnitude and distance of 
a star, we can calculate its true brightness provided it may 
be assumed that we see it undimmed by intervening fog. 
Interesting conclusions may be drawn from a dynamical 
study of the motions of the stars but it is assumed that the 
movements are not interfered with by a resisting medium. 
We calculate that in the course of time the masses of the stars 
must decrease by the loss of mass due to radiation but what 
if at the same time the stars are acquiring more mass by 
sweeping up the cosmic cloud as they pass through it? The 
cosmic cloud was thus a bogey which threatened the security 
of many of our theories of the structure and mechanism of 
the stellar universe. And so there arose discussions and 
theories of the cosmic cloud and attempts to estimate its 
probable properties. This was not speculation; it was pre- 
caution. Now that the bogey has materialised it has lost its 
frightfulness; it turns out that the cosmic cloud is so sparse 


that it is not a very serious factor in the problems I have 
mentioned, though it is perhaps not always negligible. 

I suppose it was in any case improbable that inters|ellar 
space would turn out to be entirely empty. Nature actors 
a vacuum; and we must expect individual atoms to stray 
away from stars and nebulae and get lost in the vast regions 
of space, much as dust accumulates in an empty room. We 
generally suppose that the stars have condensed out of one 
primordial nebula comprising the whole galaxy, and we can 
calculate that the condensations would not entirely drain the 
matter from the regions between them. Thus we may expect 
to find the universe a bit dusty, either by accumulation or 
because it was not properly cleaned to begin with. It is true 
that a certain amount of sweeping goes on. The stars, like 
celestial housemaids, run hither and thither, and by their 
gravitation draw in the surrounding matter. But the sweepers 
are few compared with the volume to be swept, and we can 
calculate that by this process it will take at least 10,000 billion 
years to complete the celestial spring-cleaning. 


I will come at once to the direct evidence for the existence 
of a cosmic cloud. It is well known that when light passes 
through a gas the atoms leave their characteristic mark upon 
it, so that when the light is analysed by passing it through 
a prism the spectrum shows a number of gaps or dark lines. 
These gaps, which represent the depredations of the atoms, 
indicate not only the chemical nature of the gas but how fast 
it is moving towards or away from us. For example, if we 
turn a spectroscope on to one edge of the sun we see the lines 
of a gas, e.g. iron vapour, in a position which indicates that 
the gas is coming towards us; turning it on to the other edge 
of the sun we see the same set of lines but they are now in 
a position which indicates that the gas is going away from 


us. One edge coming towards us and the other going away 
from us means that the sun is rotating a fact already dis- 
covered by watching the sunspots which appear from time 
to time on its surface. But there are a number of lines in the 
sun's spectrum which do not show this effect of rotation; 
they are seen in the same position whether we look at the 
east or the west edge of the sun. Clearly they are not formed 
in the rotating atmosphere of the sun; they must have been 
imprinted on the light after it got clear of the sun altogether. 
We have discovered a stationary gas lying somewhere 
between the sun and our telescope. Moreover we have 
discovered its chemical composition; the stationary lines 
correspond to oxygen and nitrogen. A stationary medium 
consisting of oxygen and nitrogen Why! Of course there 
15 a stationary medium consisting of oxygen and nitrogen 
between the telescope and the sun. It is only our own atmo- 
sphere we have rediscovered. 

The same method applied to the stars has, however, had 
more momentous results. The effect was first noticed by 
J. Hartmann in 1904 in 8 Orionis, which is one of the three 
stars in Orion's belt. It is a double star, but most of the light 
comes from the brighter component, and the spectrum ol 
the fainter component is not visible. We can follow the 
motion of the bright component in its orbit by observing 
the lines of its spectrum. For three days the bright com- 
ponent conies towards us and the dark lines are seen shiftec 
towards the violet; then for three days it recedes and the 
dark lines are seen shifted towards the red. This applies tc 
most of the dark lines. But there are two strong fines due 
to the element calcium, known as the H and K lines, whict 
remain in the same position all the time. Evidently these 
have a different origin from the others. They are imprintec 
on the light after it has left the moving star, and indicate 
some medium containing calcium vapour which lies betweer 
the star and our telescope. It is not the earth's atmosphere 


this time, for that does not contain calcium vapour. And in 
any case by measuring the positions of the H and K lines we 
determine the motion of the calcium vapour, and find that 
it is not connected with the earth just as we have found that 
it is not connected with the star. 

The only other "fixed line" that has been observed is the 
yellow D line of sodium. These lines of sodium and calcium, 
seen in the spectra of stars but evidently not belonging to the 
stars, have been found in the spectra of a great many stars. 
It seems a natural inference that the calcium and sodium form 
a cloud diffused through interstellar space, through which 
the light of the stars travels to reach us. This hypothesis was 
in fact proposed by Hartmann, but it was a long while before 
it became accepted. The objection was that only the very 
hottest types of stars (Types O and B) show the fixed lines. 
It was argued that if the lines were formed in a medium 
filling interstellar space all classes of stars ought to show them. 
We shall see later how this objection has been met. In the 
meantime it seemed that the high temperature of the stars 
must have something to do with the phenomenon, and 
therefore the calcium and sodium vapour must be com- 
paratively close to the star. The common belief was that it 
formed an aureole enveloping the whole double star; the 
two component stars pursued their orbits within this envelope 
without disturbing it seriously. This could be put to the test. 
So far as periodic orbital motion is concerned the calcium- 
sodium envelope need not follow the star moving to and fro 
within it; but the average motion over a long time must be 
the same for both, otherwise the star and its envelope would 
separate. This test indeed had been thoroughly applied as 
early as 1909 by V. M. Slipher, who reached conclusions 
which accord with the modern results; but his work seems 
to have been overlooked. 

In 1923 an investigation by J. S. Plaskett with the 72-inch 
reflector at the Dominion Observatory, British Columbia, 


removed all doubts on this point. Observing some forty 
stars which showed fixed lines, he found that there were 
considerable differences (sometimes very large differences) 
between the velocity of the star and the velocity of the 
calcium. Interpreted according to the foregoing view, the 
stars were leaving their haloes behind. An equally significant 
fact was that, whereas the stars had morions of their own, 
some large, some small, the calcium was always found to be 
nearly at rest in space. Not at rest relatively to the solar 
system, for the sun has an individual morion of its own; but 
relatively to the more significant standard "the mean of the 
stars", the calcium sampled in different parts of the sky was 
found to have little or no motion. This strongly suggests that 
it forms one continuous cloud. 

This is the primary evidence which leads us to picture a 
cloud of matter filling the stellar system, comparatively 
quiescent, with the stars rushing about through the midst of 
it. Light sets out from a distant star on its journey towards 
us travelling 186,000 miles every second. On and on it goes, 
year after year, with sparsely strewn atoms in and around its 
track. Now and again a calcium or a sodium atom makes 
depredations. The light has to run the gauntlet for, say, 
1000 years before it reaches the earth. It arrives depleted in 
those constituents which calcium and sodium atoms devour, 
showing therefore those gaps (dark lines) in its spectrum 
which have enabled astronomers to unravel the story. 

The longer the light journey the greater the loss by de- 
predation. Therefore the intensity the blackness of the 
stationary calcium lines should be a clue to the distance of 
the star. That was the next test to try. It was first shown to 
be fulfilled by Otto Struve. We scarcely expect this relation 
of intensity to distance to be very accurate because the cloud 
will not be uniform; the nebulae, for example, are places 
where it is strongly condensed. But smoothing out irre- 
gularities by taking the mean results for stars at different 


distances, the increase of intensity with distance is quite 
marked; moreover it increases according to the law which 
the theory of absorption would lead us to expect. 

A particular example may perhaps be more impressive 
than general statistical confirmation. It was noticed in 1910 
that the stars of high temperature in and around the con- 
stellation Perseus divide themselves into two groups according 
to their proper motions. In the foreground there is a group 
of stars, all moving across the sky in the same direction and 
apparently at the same rate, evidently forming an associated 
cluster. The apparent motion is large for this class of star, so 
that it is fairly certain that the cluster is relatively near. The 
remainder of the stars in the region show little or no apparent 
motion and form a distant background. This is a good 
opportunity for applying the test, because in observing fore- 
ground and background stars we are looking in the same 
direction through the cloud and are not so liable to be misled 
by irregularities of its density. It is found that the foreground 
and background stars can be distinguished at once by the 
intensity of the fixed calcium lines ; these show up much 
more strongly in the background stars, owing to the greater 
thickness of cloud in the way. 

A still more remarkable test has been applied by Plaskett 
and Pearce. It depends on the fact that the whole of our 
galaxy of stars is rotating about a centre far away from us 
in the direction of the constellation Ophiuchus. It is not 
rotating like a rigid body, but (as required by the law of 
gravitation) the outer parts revolve more slowly than the 
inner parts as the outer planets in the solar system revolve 
more slowly than the inner planets. By comparing the mean 
motion of the stars observed in different parts of the sky, we 
are able to detect and measure the differential motion of 
rotation. The magnitude of the effect will depend on the 
average distance of the stars surveyed; because the farther 
our survey extends, the greater will be the difference of 


velocity of the outermost and innermost stars comprised in 
it. We can use this Oort effect, as it is called, to measure the 
average distance of any class of stars, provided that the stars 
are well distributed round the sky. 

The stars which show fixed calcium lines are so remote that 
we cannot use any of the more elementary methods of 
measuring distances. But we have now two methods of 
finding the average distance of a class of stars which are 
especially appropriate to large distances, (i) by the intensity 
of the fixed calcium or sodium lines, and (2) by the Oort 
rotation effect; and we can check one against the other. 
Plaskett and Pearce first sorted out their stars into three 
groups according as the calcium lines were weak, medium 
or strong; these accordingly comprise the near, intermediate 
and distant stars. From the measured velocities they then 
determined the Oort effect, and thus found the average 
distances of the three groups. These proved to be in the order 
expected. This was the first check. 

In calculating the Oort effect the velocities shown by the 
ordinary spectral lines of the stars were used not the calcium 
lines which belong to the cloud. But Plaskett and Pearce 
also made another similar calculation using the velocities 
given by the fixed calcium lines. We have seen that the 
calcium cloud is nearly at rest relative to the mean of the 
stars, so that it evidently shares with them in the galactic 
rotation; and it should therefore show the Oort effect. 
Accordingly for each of their three groups Plaskett and 
Pearce found a distance of the cloud as well as a distance of 
the stars. Their measurement referred, of course, not to the 
whole cloud but to the part of the cloud which was per- 
forming the absorption and creating the spectral lines. If the 
veiling cloud between us and the star is uniform its average 
distance will correspond to a point half-way between us and 
the star. Thus the distance found for the cloud should always 
be half the distance of the corresponding stars. This was found 



to be closely fulfilled. The actual results which exhibit this 
are as follows : 


No. of 



km. per 


km. per 









Let us now return to the difficulty which for a long time 
baffled astronomers, namely that only certain types of stars 
show this effect. It is really due to a chapter of accidents. 
Naturally we shall only detect the absorption if there is a 
large thickness of cloud between us and the star; so that only 
stars distant more than, say, 300 parsecs are eligible. Since 
its apparent brightness must be sufficient to allow us to 
examine the spectrum, and since it must be distant more than 
300 parsecs (1000 light-years), the star must have very high 
intrinsic luminosity. That greatly restricts the possible types 
of stars. Then further the star must be of such a type that it 
does not produce the calcium and sodium lines on its own 
account; for in a stellar atmosphere these lines (if they occur) 
are strong and broad, and they may completely mask the 
fine sharp lines which the cosmic cloud superimposes. When 
both these factors are taken into consideration the limitation 
to the particular types is fully explained. 

That this explanation is right has been proved recently by 
several instances in which, owing to exceptional circum- 
stances, it has been possible to discover the fixed lines in stars 
of the "wrong" type. If the trouble is that the fixed line is 
being masked by the star's own calcium or sodium lines, it 
occasionally happens that we can surmount it. In some 


double stars the two components have extremely rapid 
motion; then at a certain phase the calcium lines of one 
component will be displaced by the motion well away to 
the right and the lines of the other component to the left, 
leaving a gap where the fixed or interstellar calcium line can 
show itself. This has duly been observed to happen. 


Why calcium and sodium? I do not for a moment suppose 
that the cloud is composed wholly or even mainly of these 
two elements. But running through the list of the elements, 
we soon satisfy ourselves that calcium and sodium are the 
only reasonably abundant elements that, under the conditions 
of stimulation prevailing in interstellar space, could yield 
spectral lines observable by us. It is no accident that the 
cosmic cloud is betrayed by three particular spectral lines, 
H, K and D ; these and no others are the lines which rarefied 
matter composed like an average sample of terrestrial matter 
would display. 

Although we can learn a great deal about the chemistry 
of the heavenly bodies we have not all the advantages that 
a laboratory analyst has. He, if he wants to find out whether 
a particular element is present in a sample of material, takes 
care to provide the conditions of heat or electrical stimulation 
which are most favourable for his investigation. We have 
to take the conditions as we find them; and if they are not 
favourable for developing a particular spectrum we miss the 
corresponding element in our search. The worst handicap of 
the astronomer is that all celestial spectra are cut off abruptly 
at about wave-length 3000 A., a point at which the laboratory 
physicist would say that spectra are just beginning to be most 
informative. We are in the position of a listener trying to 
follow a piece of music with a loud speaker that can reproduce 
only the bass notes. A layer of ozone high up in our atmo- 
ENPS 13 


sphere is opaque to radiation beyond the limit I have men- 
tioned; so we lose all the treble notes in the song of the 
celestial atoms. Calcium and sodium have deep chesty voices 
and can make themselves heard. 

Let us turn now to considerations of a more theoretical 
kind. We want to gain some idea of the density of the cosmic 
cloud. Various lines of argument prove that it must be 
extremely tenuous. One proof rests on Einstein's theory. 
For any given density there is an upper limit to the greatest 
possible extension of the cloud. For example, a globe of 
matter of the density of water cannot possibly be more than 
400 million miles in diameter. Perhaps I had better explain 
why. But the worst of explanations is that they often provoke 
more questions than they answer, and I shall not be surprised 
if you find my explanation more incredible than the state- 
ment itself. It is nevertheless a sober scientific calculation due 
originally to Schwarzschild. By Einstein's law of gravitation 
a lump of matter causes a curvature of the space which it 
occupies. If you enlarge it you add more space of the same 
curvature. You can go on enlarging it until the space has 
curved right round and closed up ; then you must perforce 
stop. That is what happens to the globe of water; when it has 
been enlarged to a diameter of nearly 400 million miles, space 
closes tightly up all round and there is nowhere to put any 
more water or anything else. Unless you have taken the 
precaution of immersing yourself in the water, you will be 

Another way of reaching the same upper limit is to con- 
sider that, if the globe is large enough, its gravitation will be 
so intense that neither light nor anything else can escape from 
it; so that it will form an entirely self-contained universe. 
In order to support myself with authority I will give a 

A luminous star of the same density as the earth, and whose 
diameter should be 250 times larger than that of the sun, would 


not, in consequence of its attraction, allow any of its rays to 
arrive at us; it is therefore possible that the largest luminous 
bodies in the universe may, through this cause, be invisible. 

Perhaps you will look on this as one more illustration of 
the disastrous effects of the Einsteinian revolution on 
respectable scientific investigation, and lament the old days 
when the teaching of Newton, Laplace and other giants of 
the past kept science in the true path of sanity. But do not 
be in too much of a hurry to blame Einstein. The passage 
quoted is from Laplace's Systime du Monde (1796). Even 
Newton thought that light might be subject to gravitation, 
and by Laplace's time it had come to be generally assumed 
that it was. Passing over a century during which (owing to 
the undulatory theory) it was generally supposed that light 
was not subject to gravitation, the first observational proof 
of the action of gravitation on light was in 1919. 

The lower the density, the larger the globe that can be 
built. Evidently we must take the density of the cosmic gas 
low enough to build a cloud which can contain our whole 
galactic system. This condition requires that the density shall 
be less than io~ l8 , that is to say one million million millionth 
of the density of water. 

A still more stringent limit is found by considering the 
observed velocities of the stars. The more gravitating matter 
there is in the stellar system, the greater are the forces to 
which the stars are subjected, and the greater will be the 
average speed of stellar motion. By this criterion it was 
found that the density of the cloud could scarcely be greater 
than io~ 2 3; but the calculation may not be very trustworthy, 
since the ideas on which it was based have been somewhat 
modified by the discovery of the rotation of our galaxy. 

These are upper limits. A more definite estimate of the 
average density of the cosmic cloud is obtained by con- 
sidering the way in which the density of a nebula tails offinto 
the normal uncondensed cloud. We shall see later that both 



the cloud and the nebulae are at a rather high temperature 
of the order 10,000 to 20,000. We can make what is 
probably a near enough guess at the average weight of the 
particles; it will be considerably greater than the corre- 
sponding quantity for the interior of a star (p. 146) because 
the atoms are less highly ionised. Then for a nebula of given 
temperature and average molecular weight it is possible to 
calculate the way the density falls off from the centre out- 
wards; and fortunately for us the density at large distances 
from the centre turns out to be nearly independent of the 
central density (which we should have been quite unable to 

There is 110 definite boundary to a nebula. The density 
continually falls off at greater and greater distances until we 
come to the outskirts of the next adjacent nebula. Thus if we 
want to know the average density of the gas in a normal 
region of space, we have to ask ourselves how far on the 
average will it be from the centre of the nearest nebula; we 
may then calculate the density as though it were part of that 
nebula. For the calculation we require, besides the distance, 
only the temperature and the average molecular weight, as 
explained above. From the observed distribution of the 
nebulae it is estimated that normally the nearest nebula is 
100 to 200 parsecs distant. This gives a density of io~ 24 , i n 
round numbers, for an average region of the cosmic cloud. 

If this is the right order of magnitude of the density, the 
amount of matter in the cloud is roughly the same as the 
amount condensed into stars. This is in agreement with a 
theoretical study of the conditions of formation of con- 
densations in a uniform primordial nebula, which indicates 
that | of the matter will form condensations and f will be 
left uncondensed. 

At the centre of a typical nebula, e.g. the Great Nebula in 
Orion, the density must be about 10,000 times greater, viz. 
icr 20 . This is one-millionth of the density in the highest 


vacuum that we can create in the laboratory. So throughout 
the present chapter I am talking about that which by 
terrestrial standards is less than nothing. 

The density will be more vivid to us if we express it in 
terms of atoms. A density of io~ 2 4 means that there is about 
one atom to the cubic centimetre, if as in the stars the 
majority of the atoms are hydrogen. I suppose it is rather 
startling to realise that in the remote solitude of interstellar 
space an atom still has neighbours within an inch of it. 
I wanted to impress on you the extreme tenuity of the cosmic 
cloud; but my last statement is likely to reverse the im- 
pression, giving you a picture of the atoms swarming as 
thickly as a plague of gnats. The picture is true enough; but 
we have to remember that an atom is a most insignificant 
quantity of matter. A moderate smoker will in the course 
of a day pollute the air with a prodigious number of atoms 
so many that, if we suppose them to diffuse evenly through 
the atmosphere all over the earth, no one will be able to 
draw a breath anywhere without inhaling a dozen atoms that 
have come from the offending pipe. 

Take a cupful of liquid, label all the atoms in it so that you 
will recognise them again, and cast it into the sea; and let the 
atoms be diffused throughout all the oceans of the earth. 
Then draw out a cupful of sea-water anywhere; it will be 
found to contain some dozens of the labelled atoms. We can 
read a literal meaning into Macbeth's words : 

Will all great Neptune's ocean wash this blood 
Clean from my hand? No, this my hand will rather 
The multitudinous seas incarnadine. 

One atom per cubic centimetre does not amount to much. 
A portion of the cosmic cloud as large as the earth could, if 
compressed, be packed in a suitcase and easily carried with 
one hand. 



The most paradoxical thing about the cosmic cloud is that 
it is intensely hot. We often speak of the intense cold of 
interstellar space. It is quite true that far away from the sun, 
at an average point in our galaxy, the temperature of any 
solid or liquid body would fall to 270 C., or 3 above 
absolute zero. That is the temperature that would be in- 
dicated by a thermometer; it is the degree of cold which the 
human body would feel, if feeling could be imagined under 
such conditions. But the diffuse cloud, by reason of its 
diffuseness, contrives to keep warm in the same conditions. 

Crossing any region of space there is a certain amount of 
heat radiated by the stars. Altogether it amounts to about 
the heat of a candle 100 yards away. You can imagine that 
it would be a bit chilly to sit out in space trying to warm 
yourself by a candle 100 yards away. If the human body 
could store up all the heat received minute by minute from 
the candle, you would in the end become warm; but matter 
is so constituted that it dissipates any heat contained in it, 
and as soon as the temperature has risen to 3 absolute this 
loss becomes sufficient to neutralise the gain. 

The reason why the diffuse cosmic gas reaches a higher 
temperature is that it has less opportunity of losing the heat 
it collects. The heat of a gas is the energy of motion of its 
particles (molecules, atoms or free electrons), and the time 
when there is a risk of losing some of this energy is during 
a collision of two particles. In air under ordinary conditions 
each particle undergoes some thousands of millions of col- 
lisions every second. In the cosmic cloud an atom encounters 
another atom about once a year; it has, however, as a milder 
excitement a collision with an electron about every five days. 
Owing to this rarity of collisions, the process of loss of heat 
which operates in ordinary solid bodies is rendered practically 
idle in the cosmic cloud. 


That, however, is not the whole secret of the high tem- 
perature of the cloud. The processes by which a body loses 
heat are closely bound up with the processes by which it 
acquires heat, so that the argument cuts both ways. The 
collisions are an opportunity for gathering in the radiant 
heat that is passing, as well as for losing it; and owing to 
their rarity the gas lets most of the radiation pass through 
without being warmed by it; that is to say, it is highly 
transparent. So if we imagine a piece of the cosmic cloud 
and a solid meteorite each sitting in front of a candle 100 yards 
away and trying to get warm, it is not immediately obvious 
which will have the advantage. The cosmic cloud secures 
very little heat but it does not easily lose what it does secure; 
the meteorite secures all that comes its way but parts with 
it easily. All we can say is that the mechanism, which 
determines what temperature the meteorite will take up, is 
practically out of action in the cosmic gas ; so that there is 
no reason for them to have the same temperature. In the 
cosmic gas the field is left clear for a secondary mechanism, 
unconnected with collisions, to take control of the tempera- 

This second mechanism is the "photo-electric effect'' 
(p. 37). A quantum of light (of sufficiently high frequency) 
falling on an atom causes an electron to shoot away at high 
speed. In interstellar space the star-light is continually 
causing this ejection of high-speed electrons. We may say 
that an electron gas at high temperature is being generated 
high temperature because of the high speeds. The electrons 
are ultimately captured again so that the electron gas is 
disappearing as fast as it is generated; but being always 
generated at high temperature it warms up the cloud. 

It is not possible here to go at all deeply into the theory; 
but the important point is that by the laws of quantum theory 
the speed of ejection of the electrons, and therefore the 
initial temperature of the electron gas, depends on the quality 


and not on the quantity of the stellar radiation. It is therefore 
the same in the depths of interstellar space as in the close 
neighbourhood of the stars; and the temperature is in fact 
not far short of the surface-temperature of the hottest stars 
responsible for the radiation. Not even the quantum theory 
provides something for nothing, and quantity must tell in 
another way. The very low intensity of star-light in inter- 
stellar space does not reduce the temperature of the electron 
gas, but it makes its generation a very slow business. The 
slowness, however, does not matter in this connection, since 
all other ways of heating or cooling the cloud have practically 
stopped and there is no competitor to outstrip. So far as we 
can estimate the cosmic cloud will take up a temperature of 
the order 15,000. 

Let us now summarise the results so far reached. We 
started with the direct observational proof that there is an 
interstellar gas which gives the H and K lines of calcium and 
the D line of sodium in the spectra of distant stars, these lines 
not being attributable to the stars themselves on account of 
the difference of motion that is indicated. We then attacked 
the problem in a different way, and by an independent 
theoretical argument concluded that there should exist inter- 
stellar matter of density about io~ 24 and temperature about 
15,000. It remains to connect the two investigations, and 
examine whether the sodium and calcium contained in a 
cloud of this density and temperature would give absorption 
lines of the intensity which we actually observe. This requires 
that we should examine the state of the atoms; for atoms give 
different spectral lines according to their state of ionisation. 
For example, calcium is a divalent element with two rather 
loosely attached electrons. Under the conditions above stated 
we find that the great majority of the calcium atoms will be 
without these two electrons; they have gone off to form part 
of the electron gas to which I have referred. 

Now the calcium atom with two electrons missing gives 


no observable spectrum; it is not these atoms that we are 
concerned with. The H and K lines are produced by calcium 
atoms with one electron missing. About i in 800 of the 
calcium atoms will be in this state. Complete calcium atoms 
are very rare in the cloud about i in 50,000,000; the rarity 
explains why we do not observe in the spectrum of the cloud 
the lines of un-ionised calcium. 

Calcium is a fairly abundant element forming about 
i per cent, of the whole mass of the earth. If we allow the 
same proportion in the cloud, and remember that only 
8 o^ of the atoms are in a state to cause absorption of H and 
K lines, we find that there is about one active calcium atom 
in a cubic yard of cosmic cloud. Consider now a star 
1000 light-years away. We see it across a screen 1000 light- 
years thick containing one absorbing atom per cubic yard. 
What intensity of absorption line will such a screen produce? 
The physicist is able to answer this question from his experi- 
mental and theoretical knowledge; and when we compare 
his calculation with the intensity (width and blackness) of 
the fixed H and K lines that we actually observe in a star 
1000 light-years away, the agreement is as close as could be 

Unfortunately this agreement for calcium is marred by a 
complete disagreement for sodium which we are unable to 
explain. The D line is produced by the complete sodium atom; 
but in the conditions that we have calculated for the cloud 
the sodium ought to be nearly all ionised, and complete 
atoms should be far too rare to give the absorption lines that 
we observe. Even if the cloud consisted entirely of sodium 
there would still not be enough complete sodium atoms. 
And so I have to leave my story without a happy ending. 
But perhaps after all it is a happy ending that stimulates us 
to pursue farther our investigations because there is still 
something fundamental to be found out. 


Although the cosmic cloud is generally invisible there are 
denser patches which are faintly luminous. These are the 
gaseous nebulae of which an example is shown in Plate 2. 
The most interesting part of the study of gaseous nebulae 
deals with the origin and nature of their light. 

We are familiar with bodies such as the sun which shine 
by their own light, and with bodies such as the moon which 
shine by borrowed light. A gaseous nebula is in a sense 
intermediate. The nebula is dependent for its light on the 
stars which He in the midst of it; but it does not simply reflect 
their light; their radiation falls on the atoms of the nebula 
and stimulates them so that they emit light of a different 
kind. To use the recognised term for this process, the nebula 
is fluorescent. It is only the stars of very high temperature 
that can cause a nebula to shine ; the sun would not be capable. 
So even the densest portions of the cosmic cloud will remain 
dark unless there are high-temperature stars in the neigh- 
bourhood. We may suspect that the dark obscuring nebulae 
(Plate 3) are similar to the luminous nebulae but lack the 
stimulating stars. It is, however, very difficult to account for 
their opacity if they consist of gas alone; and for that reason 
astronomers nowadays usually look on them as clouds of 
dust or meteoric matter. Whatever be the solution, there is 
an intimate association between the obscuring nebulae and 
the luminous nebulae; for we often see in the same nebula 
luminous portions which grade continuously into dark 
obscuring portions. 

Keeping to the luminous nebulae, their spectrum is, as we 
should expect, that of a practically transparent layer of gas; 
that is to say, it is a spectrum consisting of bright lines. The 
spectrum of hydrogen is a prominent, but by no means the 
most prominent, feature of the spectrum. Ionised helium, 
i.e. atoms of helium which have lost one satellite electron, 


Mount Wilson Observatory 
The Horse's Head in Orion 


can also be recognised. The rest of the spectrum consists of 
lines entirely unknown in the laboratory. Most of the visible 
light comes from two green lines whose source has been 
named nebulium. The photographic light contains another 
very prominent line whose source has not been specially 
named. Named or not, the light of the nebulae is for the 
most part like nothing on earth. 

The modern theory of the sequence of the atomic numbers 
of the elements (p. 30) leaves no room for new elements 
until we reach very high atomic numbers. Up to the point 
where the first gap occurs, the physicist would be almost as 
surprised to discover a new element as the mathematician to 
discover a new integer. Thus for many years astronomers 
have been convinced that nebulium is an alias of some very 
familiar element, and that the sources of the other unknown 
lines are likewise familiar substances. The problem was how 
to force some familiar element to emit the strange light which 
it is so reluctant to give in the laboratory. Laboratory treat- 
ment of atoms is still somewhat crude. Our method of 
making an atom work is to knock it about; and if it does not 
do what we want, knock it still harder. 

But is it likely that this treatment will bring out light of 
the kind emitted in the nebulae? In the nebulae the atoms 
have a very quiet life. We have seen that in the cosmic cloud 
the only break of monotony is an encounter with an electron 
about once a week. In the thousand-fold denser nebula things 
are speeded up proportionately; but even so, to an atom 
whose natural periodicity is of the order io~9 seconds, the 
interval between encounters must seem almost an eternity. 

Let us look at it another way. We can measure roughly 
the amount of light emitted by a luminous nebula, e.g. the 
Orion Nebula, and we can express the result as a certain 
number of quanta (or photons) emitted per second. We 
know also the size of the nebula, and the theory described 
in this chapter has given a general idea of the density. We 


can therefore estimate the total number of atoms in the 
nebula. Hence we can calculate how often on the average 
each atom is called upon to emit a photon. We find that its 
turn comes round about once a century. 

The secret of nebulium was discovered by I. S. Bowen in 
1927. He found that the strange light was due to what are 
known theoretically as "forbidden transitions". We have 
seen that there are a number of possible orbits for a satellite 
electron, and that light is emitted when the electron jumps 
from an orbit of higher energy to an orbit of lower energy. 
But the electron does not jump indiscriminately. It is as 
though the orbits were connected by cross-passages; some 
pairs of orbits have a cross-passage, others have not. It may 
happen, for example, that an electron in orbit No. 3 can drop 
to No. 2 or to a still lower orbit No. i, but it will not drop 
from No. 2 to No. i. In that case the passage from No. 2 to 
No. i is called a forbidden transition. The theory of the atom 
has furnished us with rules that determine which transitions 
are forbidden. 

It was realised that the transitions are only relatively 
forbidden. The electron in the above example can drop from 
orbit No. 2 to No. i, only the chance of its doing so in any 
reasonable time is small; and if it does not act quickly, it will 
be whisked out of orbit No. 2 by the collisions and absorp- 
tions that are continually occurring in terrestrial conditions. 
Bowen realised that in a nebula an electron, which had been 
knocked up into orbit No. 2 with only the orbit No. i below 
it, would ultimately have to make the forbidden transition. 
There being nothing to disturb or release it, it would remain 
a prisoner in orbit Mo. 2 until its obstinacy gave out. The 
unfamiliar lines in the nebular spectrum correspond to for- 
bidden transitions, and for that reason they are only emitted 
in extremely quiescent conditions such as prevail in a nebula. 

The proof lies in the identification of the lines. Nebulium 
is doubly ionised oxygen. All the other conspicuous lines in 


the nebulae whose origin was previously unknown are for- 
bidden lines either of singly or doubly ionised oxygen or of 
singly ionised nitrogen. 

How do we know what are the forbidden lines of oxygen 
if we cannot ourselves persuade oxygen to produce them? 
Except in one specially simple case we cannot calculate the 
spectrum of an atom by pure theory, and we are reduced to 
measuring wave-lengths experimentally. But there is no 
need to measure the wave-lengths of all the lines in the 
spectrum; when we have measured a certain number, we 
can calculate the rest. The rule of calculation, which is a 
simple one, is well known in quantum theory. If the cal- 
culated line is not forbidden, we can observe it and so verify 
the rule; but the same rule also enables us to calculate the 
wave-lengths of the forbidden lines which we cannot observe. 

Thus Bowen recognised the nebulium spectrum as a 
spectrum of oxygen although terrestrial oxygen had never 
been known to produce it. He recognised it as the spectrum, 
known in theory, which under ordinary circumstances 
oxygen is forbidden to produce. It is still unproduced in the 
laboratory. Even if we could secure sufficient quietude for 
the atoms, we could scarcely expect to detect the light. For 
the atoms, as we have seen, take their time over emitting it 
and cannot be made to work faster than, say, once a second. 
That means that the light is generated very feebly, and a 
source of astronomical dimensions is required to yield an 
appreciable quantity. 

So the source of 

The light that never was. on sea or land 

is a familiar enough substance. It is oxygen and nitrogen 
or, if you like, common air. 


Or if they list to try 
Conjecture, he his fabric of the heavens 
Hath left to their disputes. MILTON, Paradise Lost. 


THIS chapter describes a material system on the largest scale 
yet imagined, namely the system of the galaxies. Let us first 
understand what a galaxy is. The following is a recipe 
for making galaxies: Take about ten thousand million stars. 
Spread them so that on the average light takes three or four 
years to pass from one to the next. Add about the same 
amount of matter in the form of diffuse gas between the stars. 
Roll it all out flat. Set it spinning in its own plane. Then you 
will obtain an object which, viewed from a sufficient distance, 
will probably look more or less like the spiral nebula shown 
in Plate 4. 

The evidence is now considered conclusive that the spiral 
nebulae (not to be confused with the gaseous nebulae con- 
sidered in the preceding chapter) are immense systems of 
stars. They are presumably the units that we have to deal 
with in a survey of the universe as a whole. We, of course, 
live in one of these galaxies, of which the sun and all ordinarily 
recognised stars are members. We call it the Galaxy. Being 
inside it we do not get so good a general view of it, and it is 
difficult to compare it with the other galaxies. From an 
observational point of view there is some doubt whether our 
galaxy is a normal specimen; it appears to be outstandingly 
large. But the tendency of recent investigations has been to 
level up things, and make our galaxy seem less abnormal. 


M. 51 in Canes Venatici 


Judging by sample counts in different parts of the sky there 
are some millions of these galaxies visible with our largest 
telescopes; and goodness knows how many there are beyond 
their range. Or rather I think I also know more or less. 
But that is all theory, and you will have enough of that later 

The spiral nebulae or galaxies will now be our units. Just 
as the chemist generally takes an atom as his unit and does 
not need to trouble about anything smaller, dealing as he 
does with aggregations of myriads of atoms, so we shall take 
the galaxies as our atoms, not recognising anything smaller, 
and discuss an aggregation of, say, a billion galaxies which, 
it seems, constitutes the universe. 

Let us now see where we have got to in the scale of size: 


Distance of sun 93 ,000,000 

Limit of solar system (Orbit of Pluto) 3,600,000,000 

Distance of nearest star 25 ,000,000,000,000 

Distance of nearest galaxy 6,000,000,000,000,000,000 

Original circumference of 

the universe 40,000,000,000,000,000,000,000 

The last entry is here rather premature, but we shall refer to 
it later (p. 218). 

I have sometimes heard complaints that astronomical 
numbers are too large to comprehend. I must confess that 
the numbers in the present chapter are rather out of the 
ordinary. But I cannot see why anyone should find a diffi- 
culty with the numbers occurring in the lesser astronomical 
systems. They are just the sort of figures with which our 
economists and politicians are always dealing, quoted any day 
in the newspapers. People say that they cannot realise these 
big numbers. But that is the last thing anyone wants to do 
with big numbers to realise them. Do you suppose that, 
as Budget Day approaches, the Chancellor of the Exchequer 


throws himself into a state of trance in which he can visualise 
and gloat over 800,000,000 sovereigns, or notes, or com- 
modity values, or whatever they are, that he is about to 
amass? His only concern is to make quite sure that, although 
neither 800,000,000 nor 8,000,000,000 is a number that can 
possibly be * ' realised ' ', he does not forget which is which. The 
purpose of the foregoing table of distances is that we may 
keep the different scales distinct in our minds; in particular 
we have to note that it is a very big step up in scale when 
we pass from a system of stars to a system of galaxies. 

How do we find the distances of the galaxies? For a start, 
we can actually photograph some of the brightest of the stars 
in the nearest galaxies. Their distance makes them appear 
very faint; and the faintness is a measure of the distance, 
provided that it is known how bright these same stars would 
appear at a standard distance, and provided also that there is 
no intervening fog. If all motor cars were equipped with 
lamps of standard power, it would be possible to tell how 
far off each car was by carefully measuring the apparent 
brightness of its lamps. The galaxies carry many lamps, and 
among these we can recognise some that are known to be 
of standard power, namely the Cepheid Variables. If we 
observe in one of the galaxies a star varying in the Cepheid 
manner with a period of 10 days, we look up in our list the 
standard light-power of a Cepheid of 10 days' period (as 
determined from the measurements of such stars in our own 
neighbourhood), and the distance of the galaxy is then 
immediately deduced. E. P. Hubble has measured the dis- 
tances of a few of the nearest galaxies in this way. For greater 
distances less satisfactory methods are employed, and I daresay 
the distances assigned to the remoter galaxies are a bit doubt- 
ful. However, we think that they give a reasonably good idea 
of the system. 

We can also determine how fast a galaxy is moving 
towards or away from us in the line of sight by measuring 


the Doppler shift of the lines in its spectrum a method 
applicable to all luminous objects whose spectra are not 
entirely featureless. For the stars of our own system it is rare 
to find velocities above 100 miles per second ; but the velocities 
of the external galaxies are generally much larger, amounting 
to hundreds or thousands of miles per second. 

There is a remarkable feature about these motions. The 
more distant the galaxy, the faster its motion. Moreover, the 
galaxies are almost unanimously running away from us. 

The nearest of the external galaxies is about a million 
light-years away. At present the farthest limit of our survey 
is about 150 million light-years. There is a spiral nebula in 
the constellation Gemini whose distance is such that the light 
waves had to start 150 million years ago in order to reach us 
to-day; this nebula is running away from us at 15,000 miles 
a second. Intermediate galaxies recede at less speed, the speed 
being, as nearly as we can tell, proportional to the distance. 
The system undoubtedly extends farther than 150 million 
light-years; but the more remote galaxies are so faint that it 
has not yet been practicable to make the measurements 
necessary to determine their speeds. [Announcement has just 
been made (November, 1934) of a still more remote nebula 
in Bootes receding at 24,300 miles a second.] 

There are five exceptions to the rule that the external 
galaxies are receding from us; but the exceptional behaviour 
is confined to galaxies in our immediate neighbourhood and 
is probably not of real importance. By the law of propor- 
tionality of speed to distance the closest galaxies, distant from 
one to two million light-years, should have an outward 
speed from 100 to 200 miles per second. This is scarcely large 
enough to predominate over various accidental causes which 
may be operating; so that no great attention need be paid 
to an occasional reversal in this region. As soon as the distance 
effect becomes too large to be masked, the recession of the 
galaxies is unanimous. 



If all the galaxies are going away and none are moving 
inwards, there will come a time when the region that we 
now survey is vacated. I cannot say that that will make much 
difference to earthly affairs ; but the astronomers of that future 
time will lose one of the most fascinating and beautiful 
features of the heavens. Nor is that date so very far off. I do 
not want to be unduly alarmist; but the nebulae double their 
distances from us every 1300 million years, and astronomers 
will have to double the apertures of their largest telescopes 
every 1300 million years merely to keep up with their 
recession. Seriously, 1300 million years cannot be looked 
upon as a long period of cosmic history; it is about the age 
assigned to the oldest terrestrial rocks; and it is certainly a 
novel idea that anything much can have happened to the vast 
system of the universe within geological times. It implies 
that the time-scale of change and evolution is much shorter 
than we were inclined to think a few years ago. We have 
to speed up the evolution of the stars in order to harmonise 
with it. 

The running away of the galaxies does not mean that they 
have a kind of aversion from us. They are not avoiding our 
own galaxy; it is not important enough for that. If we con- 
sider carefully the observational law, that the speed of re- 
cession is proportional to the distance, we see that the galaxies 
are separating away from one another just as much as they 
are separating away from our galaxy. An even dilatation of 
the whole system is occurring. If this lecture-room were to 
expand to twice its present size, the seats all separating from 
one another in proportion, you would at first think that 
everyone was moving away from you. But everyone else 
would be having the same experience. It is that kind of 
expansion which is occurring in the system of the galaxies. 
Since the rate of mutual recession, or rate of increase of 


distance from any galaxy to any other, is proportional to the 
distance, all distances take the same time to become doubled, 
viz. 1300 million years. 

So the system of the galaxies is expanding as a gas expands, 
its atoms getting farther and farther apart from one another. 
(You will remember that in this super-physics of the universe 
the galaxies are our indivisible units or atoms.) If the astro- 
nomers are right, it is a straightforward conclusion from the 
observational measurements that the system of the galaxies 
is expanding or, since the system of the galaxies is all the 
universe we know that the universe is expanding. There is 
no subtlety or metaphysics about it. Except that the system 
concerned is of unaccustomedly large dimensions, it is easy 
enough to apprehend. 

But are we sure of our observational facts? Scientific men 
are rather fond of saying pontifically that one ought to be 
quite sure of one's observational facts before embarking on 
theory. Fortunately those who give this advice do not 
practise what they preach. Observation and theory get on 
best when they are mixed together, both helping one another 
in the pursuit of truth. It is a good rule not to put overmuch 
confidence in a theory until it has been confirmed by 
observation. I hope I shall not shock the experimental 
physicists too much if I add that it is also a good rule not to 
put overmuch confidence in the observational results that are 
put forward until they have been confirmed by theory. 

So in starting to theorise about the expanding universe I am 
not taking it for granted that the observational evidence 
which we have been considering is entirely certain. That is 
what I want to find out whether theory confirms it. At the 
start we have a certain reluctance to accept these observational 
results at their face value. It is not the expansion of the 
universe but the rapid expansion which makes us look at these 
observational results very critically; for if they are true they 
play havoc with our former ideas of the time-scale of 



evolutionary development. If the speeds found for the 
spiral nebulae are genuine, there is no escape from this rapid 

But are they genuine? It is scarcely true to say that we 
observe these velocities of recession. We observe a shift of the 
spectrum to the red; and although such a shift is usually due 
to recession of the object, it is not inconceivable that it should 
sometimes arise from another cause. I can only say that 
nothing in our knowledge of physics as it stands to-day gives 
any hint of an alternative cause for the red-shift of the 
nebular spectra; there would have to be some profound 
modification either in the theory of light or in astronomical 
conclusions generally. It can be objected that our knowledge 
is incomplete and that there is a possibility of unforeseen 
developments; but that might be urged against most of our 
scientific conclusions, and it is misleading to remember it 
suddenly in one particular connection. In a recent General 
Catalogue of Radial Velocities the determinations for all other 
objects are given in a column headed " Radial Velocity", but 
for the spiral nebulae the column is headed "Apparent Radial 
Velocity". To many that will seem commendable caution; 
but to me it seems like the caution of the minister who wrote 
to his wife "I shall be home (D.V.) on Friday; and in any 
case by Saturday", 

In introducing theory, I must emphasise that it was theory 
that first suggested a systematic motion of recession of the 
spiral nebulae and so led to a search for this effect. The 
theoretical possibility was first discovered by W. de Sitter 
in 1917. Only three radial velocities of nebulae were known 
at that time, and they somewhat lamely supported his theory 
by a majority of 2 to i. Since then it has been possible to 
investigate the more remote nebulae whose support is so far 
unanimous; this progress has been mainly due to V. M. 
Slipher at the Lowell Observatory and to M. L. Humason 
at Mount Wilson Observatory. The linear law of propor- 


tionality between speed and distance was found by E. H. 
Hubble. Meanwhile the theory has also developed, and it 
has taken the form especially associated with the names of 
A. Friedman and G. Lemaitre. 

The theory of relativity predicts the existence of a certain 
force which we call cosmical repulsion. It is directly propor- 
tional to the distance of the object concerned. It is so weak 
that we can leave it out of account in discussing the motions 
of the planets round the sun or indeed any morion within 
the limits of our own galaxy. But since it increases pro- 
portionately to the distance we shall, if we go far enough, 
find it significant. Will it have become significant at the 
distance of the spiral nebulae? The theory of relativity could 
not say; it did not predict the magnitude of the force. It 
could only suggest that a search be made as to the motions 
of these remote objects; and if a general running away, such 
as would be produced by a repulsive force, were discovered, 
it might well be the manifestation of this cosmical repulsion. 

In the foregoing paragraph I have said that the repulsion 
is proportional to the distance of the object. Distance from 
what? From anywhere you like. We take it to be distance 
from the earth, or rather from our galaxy since the galaxies 
are our indivisible atoms. But an observer in another galaxy 
can take it to be distance from him. It does not matter; we 
shall all obtain the same results so far as anything observable 
is concerned. Cosmical repulsion is a dispersing force tending 
to make a system expand uniformly not diverging from 
any centre in particular, but such that all internal distances 
increase at the same rate. That corresponds precisely to the 
kind of expansion we observe in the system of the galaxies. 

I have said that relativity theory predicts a force of cosmical 
repulsion. When using its own technical language, relativity 
theory does not talk about anything so crude as force; it 
describes the phenomena by means of curvature of space- 
time. But for practical purposes the curvature of space-time 


involved in gravitational effects is very nearly equivalent to 
the Newtonian force of gravitation; and the force of cosmical 
repulsion is similarly a translation into Newtonian language 
of another curvature effect demanded by relativity theory. 
These translations must, of course, be used with caution and 
not pressed to apply in extreme circumstances. There would 
be no object in the recondite phraseology of the theory if a 
familiar translation were equally satisfactory for all purposes. 
However, the actual relativity effect is represented with 
sufficient accuracy by a force of cosmical repulsion at any 
-ate up to the greatest distances that we actually observe. 

Cosmical repulsion is not the only force at work. The 
^alaxies exert on one another their ordinary gravitational 
attraction approximately according. to Newton's law. This 
makes them tend to cling together. So we really have a 
contest of two forces, Newtonian attraction trying to keep 
the universe together and cosmical repulsion trying to scatter 
it. If our theory is right cosmical repulsion must have got 
the upper hand, because the galaxies are actually being 
scattered. Having got the advantage, cosmical repulsion will 
keep it; because, as the nebulae become farther apart, their 
mutual attraction will become weaker and offer less opposi- 
tion to the scattering force. 

In connection with cosmical repulsion we define an im- 
portant constant of nature, called the cosmical constant, i.e. the 
amount of the cosmical repulsion at unit distance from the 
observer. This constant is generally denoted by A. Ordinary 
relativity theory does not foretell the magnitude of A, or even 
its sign (plus or minus). All that it insists is that A is not zero ; 
for the theory would then cease to be a relativity theory.* 
It was a defect of Einstein's original theory, first remedied 

* The cosmical constant expresses a relation of scale between two types 
of phenomena. So long as it is expressed by any number, however small, 
the relation remains recognised. But if it is expressed by zero the relation 
is broken. 


by H. Weyl, that it implied the existence of an absolute 
standard of length a conception as foreign to the relativistic 
point of view as absolute motion, absolute simultaneity, 
absolute rotation, etc. To set A=o implies a reversion to the 
imperfectly relativistic theory a step which is no more to 
be thought of than a return to the Newtonian theory. 

Accordingly the adopted value of the cosmical constant is 
generally determined from the observed rate of recession of 
the galaxies. Such a determination is necessarily provisional 
since it assumes that the large receding velocities are due to 
cosmical repulsion and not to other causes. We shall see later, 
however, that the theory of the cosmical constant can be 
approached in another way which gives a definite deter- 
mination of its value independent of the astronomical 
evidence, and thereby provides a check on the whole theory. 


The reader who has followed articles and discussions on this 
subject may have wished to interrupt me with a question. 
"You have been describing the expansion of a big system of 
galaxies which forms the material universe; but is not the 
'expansion of the universe' understood to mean something 
more than this not just the pushing farther back of the 
boundaries of a material system, but an expansion, an 
inflation, of space itself?" That is true, and I must say a little 
about the expansion of space although the idea is more 
difficult to follow. 

A new phenomenon is naturally considered and described 
in relation to the general physical theories prevailing at the 
time. Thus a new atomic phenomenon would nowadays be 
described according to wave mechanics, although it might 
happen to have little or no concern with the distinctive features 
of wave mechanics, and a classical description and explanation 
would be adequate. Although it is useful to recognise that 


the phenomenon does not take us beyond the limits of 
classical theory, we shall not get full value out of it as a 
contribution to the general development of science unless we 
weave it into the most up-to-date point of view. The 
physicists of a hundred years ago might well be surprised to 
learn that the positron is considered to require an intricate 
explanation barely comprehensible to anyone but a mathe- 
matician, and that it is sometimes even claimed to be a 
confirmation of modern ideas; they would see in it rather 
a confirmation of their own commonsense view that there 
are two electric fluids with perfectly symmetrical properties 
capable of cancelling one another. In the same way we have 
here encountered a new astronomical phenomenon which, 
on the face of it, is nothing more than an ordinary expansion 
of a material system. Modern theory is only involved to the 
extent of suggesting the cause of scattering (cosmical re- 
pulsion), which otherwise has to be postulated ad hoc. The 
reason why we do not rest content with this description is 
that the system is on a very much larger scale than any system 
whose expansion has previously been studied; and the large 
scale brings out certain differences between the modern 
scientific outlook and the classical outlook which would be 
insignificant in a smaller system. It is in this connection that 
the idea of an expansion of space occurs. In a briefer account 
of the expanding universe it would, I think, be justifiable to 
omit all reference to expanding space; just as in briefly 
introducing the positron I have not referred to Dirac's theory 
of it as an occasional vacancy in an infinitude of occupied 
negative energy-levels. 

We have seen that the speed of recession of a galaxy is 
proportional to its distance. At 150 million light-years the 
speed is 15,000 miles a second; at 1500 million light-years 
the speed should be 150,000 miles a second. But we cannot 
go on indefinitely like that. At 1900 million light-years we 
get 190,000 miles a second, which is greater than the speed 


of light; so that we are obviously heading for trouble. The 
trouble indeed is so near that if Dr Hubble had been armed 
with a looo-inch telescope instead of a loo-inch he would 
probably have landed us in it already. 

Einstein about 1916 seems to have had a premonition that 
if we include very great distances in our scheme of things we 
are asking for trouble. That queer quantity "infinity" is the 
very mischief, and no rational physicist should have anything 
to do with it. Perhaps that is why mathematicians represent 
it by a sign like a love-knot. Einstein therefore adopted a 
type of space in which there are no distances beyond a certain 
amount, just as on the earth's surface there are no distances 
greater than 12,000 miles. We have just seen that there is 
trouble in store for us if we go out to too great a distance 
in the system of the galaxies; but Einstein has taken the 
precaution of closing up the universe so that we cannot 
wander too far. 

It is not an accident that the closure of space saves the 
situation. We have seen that the force of cosmical repulsion 
is, like the force of gravity, an approximate equivalent in 
familiar language of the curvature of space-time in relativity 
theory. The closing up of space, so that its volume is finite 
and distances cannot exceed a finite limit, also results from 
the curvature. Thus the extent of space and the magnitude 
of the force of cosmical repulsion are proportioned to one 
another, both depending on the same cosmical constant; and 
their relation is such that the anticipated trouble cannot arise. 

To sum up : if we accept the force of cosmical repulsion 
offered by relativity theory, we should for consistency accept 
the finitude of space that goes along with it on that theory. 
It is quite true that the latter scarcely affects the problem with 
which we are primarily engaged, viz. the expansion of the 
system of the galaxies, unless we contemplate distances ten 
times greater than those yet observed; and we have as yet 
no evidence that the system extends so far as that. But when 


we go on to consider the structure and evolution of the 
universe as a whole, we naturally appeal to the complete 
self-consistent theory. 

If the system of the galaxies extends throughout closed 
finite space, it can only expand if the space itself expands. 
That is how we are led to contemplate expanding space as 
well as an expanding material system. 


To simplify things we shall suppose that the distribution of 
the galaxies is uniform throughout space. Space will then 
be spherical; that is to say, it will be like the surface of a 
sphere, only with one more dimension which you must 
imagine as best you can. More technically it is like the three- 
dimensional surface or boundary of a hypersphere in four 
dimensions. I say it is like the surface of a hypersphere that 
we can make the simplest kind of map of space by drawing 
it on a hypersphere. I do not say it is the surface of a hyper- 
sphere, for the hypersphere is only the scaffolding of the 
map. Since the idea of the map is that the whole external 
world corresponds to the surface of the hypersphere, the 
interior and exterior of the hypersphere can have no objective 

A being limited to the surface of a sphere will, if he goes 
straight ahead turning neither to the right nor to the left, 
ultimately find himself back at his starting point. Similarly 
you, limited to a three-dimensional space which is like the 
surface of a hypersphere, will, if you go straight ahead, arrive 
back at your starting point. I cannot say exactly how far you 
will have to go, but the distance is not less than 6000 million 
light-years; it may be five or ten times as much, but I think 
not more. Only you had better hurry up, because the 
universe is expanding, and the longer you put it off the 
farther you will have to go. As a matter of fact it is too late 


to start now even if you travel with the speed of light. 
Adopting my minimum figure of 6000 million light-years, 
it will take you 1500 million years to go a quarter way 
round. But we have seen that the expansion is such that 
distances are doubled in 1300 million years. So that the 
remaining three-quarters of your circuit, instead of being 
4500 million light-years, will now have become more than 
9000 million light-years. You are farther off than when you 
started. One is reminded of the effort of Alice and the Red 

"Well, in our country," said Alice, still panting a little, " you'd 
generally get to somewhere else if you ran very fast for a long 
time, as we've been doing." 

"A slow sort of country!" said the Queen. "Now here, you 
see, it takes all the running you can do, to keep in the same place. 
If you want to get somewhere else, you must run at least twice 
as fast as that." 

Spherical space presents many such curiosities, but we 
shall not here linger over them.* For the most part they do 
not lead to anything that could come under practical observa- 

/ In contemplating the dispersing system of the galaxies we 
cannot refrain from asking, What has it come from? Where 
is it going to ? To the latter question there is, so far as we can 
see, only one answer. The system will go on dispersing for 
ever the galaxies scattering more and more widely. 
Cosmical repulsion increases the distances between the 
galaxies, but it does not make an individual galaxy grow 
any larger. This is because the dispersal of a system only 
occurs if the repulsion exceeds the countervailing gravita- 
tional attraction of the parts of the system. In the galaxies 
and other smaller systems gravitational attraction always 
predominates. So although we are parting company with 
the millions of galaxies around us, we shall keep with us a 
* See The Expanding Universe, Ch. HI. 


galaxy of some 10,000 million stars, which a few years ago 
would have been considered a fairly commodious universe. 

It is more difficult to decide what the universe started from. 
For my part I choose the hypothesis which provides the 
most quiescent and orderly beginning of things. If you 
prefer the view (favoured by Lemaitre) that the universe 
started with the thunder of an explosion, there is nothing in 
our present knowledge to gainsay you ; only it seems inartistic 
to give a universe, built to contain a natural cause of ex- 
pansion, an additional shove off at the start. 

We have seen that Newtonian attraction and cosmical 
repulsion are two opposing forces. It would seem that in the 
initial state of things these two forces just balanced, so that 
ideally the universe might have remained in this embryo 
state for untold ages. But it can be shown that the equili- 
brium is unstable. If cosmical repulsion once gets the upper 
hand it will keep it (p. 214), and the universe will go on 
expanding ; similarly if Newtonian attraction gets the upper 
hand it will keep it, and the universe will go on contracting. 
Sooner or later some slight disturbance of perfect equilibrium 
was bound to occur and cause the universe to topple off its 
balance one way or the other. Several investigators have 
tried to examine whether there was some definite cause 
deciding that the universe should fall into a state of expansion 
rather than contraction; but no very decisive conclusion has 
been reached. 

According to observation the speed of recession of the 
spiral nebulae is (in round numbers) 500 km. per sec. per 
megaparsec;* that is to say, those at i megaparsec distance 
recede at 500 km. per sec., those at 10 megaparsecs distance 
recede at 5000 km. per sec., and so on. From this datum we 
can by Lemaitre's theory evaluate the cosmical constant and 
several important characteristics of the universe. Assuming 
that the universe started from the state of balance described 
* i megaparsec3'26 million light-years. 


above, its initial radius was about 1000 million light-years. 
It has since expanded; but the present radius can only be 
found by introducing very precarious estimates of the average 
density of matter in the system of the galaxies. 

We also deduce by Lemaitre's theory that the total amount 
of matter in the universe is about io w times the sun's mass. 
If an average galaxy contains 10,000 million stars, this would 
provide for about a billion (io 12 ) galaxies. Another form of 
the result is that there are io?9 protons and as many electrons 
in the universe. That is quite a useful thing to know. We 
shall find confirmation of this number in Chapter XL 

I am told that this is not the first attempt to compute the 
number of particles in the universe. There is an earlier 
calculation by Archimedes.* 

There are some, king Gelon, who think that the number of 
the sand is infinite in multitude; and I mean by the sand not only 
that which exists about Syracuse and the rest of Sicily but also 
that which is found in every region inhabited or uninhabited. 
Again there are some who, without regarding it as infinite, yet 
think that no number has been named which is great enough to 

exceed its multitude But I will try to show you by means of 

geometrical proofs, which you will be able to follow, that, of 
the numbers named by me, and given in the work which I sent 
to Zeuxippus, some exceed not only the number of the mass of 
sand equal in magnitude to the earth, but also that of a mass equal 
in magnitude to the universe. 

The calculation proceeds by steps, from sand-grains to 
poppy-seeds, to finger-breadths, to stadia, to the diameter 
of the earth, to the diameter of the universe according to 
"the common account, as you have heard from astronomers ", 
and finally to the many times greater universe recently 
advocated by Aristarchus. Archimedes concludes that 
"a sphere of the size attributed by Aristarchus to the sphere 

* Sir Thomas Heath, The Works of Archimedes, pp. 221-232. 


of the fixed stars would contain a number of grains of sand 
less than io 6 3". 

I will not enter into controversy with my venerable rival. 
I feel that we are drawn together by his concluding remark 

I conceive that these things, king Gelon, will appear incredible 
to the great majority of people who have not studied mathe- 

The conception of the expanding universe seems to crown 
the edifice of physical science like a lofty pinnacle. Or perhaps 
its strange fantastic character suggests that it would be more 
aptly compared to a gargoyle. But for my part I do not look 
on it either as a pinnacle or a gargoyle. I believe that it is one 
of the main pillars of the edifice. 

The cosmical constant is the agent behind the phenomenon 
of the recession of the galaxies. But it is also the agent behind 
a great deal more. A few years ago I became strongly 
convinced that in these astronomical discoveries in the re- 
moteness of space we had picked up the key to the mysteries 
of the proton and electron. All that I have since been able 
to work out confirms my conviction. In spherical space 
those who start off in one direction must ultimately meet 
those who started off in the opposite direction; so in science 
astronomers who went in search of the inconceivably great 
are now meeting atomic physicists who went in search of 
the inconceivably small. 

The same cosmical constant found from the motions of the 
galaxies can also be found from the properties of electrons 
and protons studied in the laboratory. We have thus two 
independent determinations of the cosmical constant which 
are found to check one another as closely as could be 
expected. The theory of the laboratory determination the 
formula giving A in terms of the other better known constants 


of Nature will be treated seriously in Chapter xi. Here I 
introduce only preliminary considerations. 

When we assert that the universe expands, what is our 
standard of constancy ? There is no particular subtlety about 
the answer; the expansion is relative to the standards that 
we ordinarily employ. It is relative to the standard metre 
bar, for example, or to the wave-length of cadmium light 
which is often suggested as a more ideal standard, or to any 
of the linear dimensions associated with atoms, electrons, etc. 
which are regarded as "natural constants" in atomic physics. 
But if the universe is expanding relatively to these standards, 
all these standards are shrinking relatively to the universe. 
The theory of the expanding universe is also the theory of the 
shrinking atom. Thus we cannot detach the theory of the 
universe from the theory of the atom. We must not think 
of the cosmical constant as an agent which manifests itself 
only in the super-system of the galaxies and is insignificant 
in the atom and other small-scale systems. It manifests itself 
in a relation (of size) between the super-system of the 
galaxies and small-scale systems, and it is no more a charac- 
teristic of one end of the relation than of the other. Thus we 
ought to be able to approach the cosmical constant through 
the theory of the atom (or more explicitly through those 
equations of quantum theory which determine the extension 
of small-scale systems) as well as through the theory of the 

According to the principle of relativity we can only 
observe and have knowledge of the relations of things. So 
when we refer to the properties of any object we must always 
have a comparison object in mind. If we speak of its velocity, 
we mean its velocity relative to some comparison object or 
set of landmarks. If we speak of its size we must have some 
standard extension to compare it with. Imagine yourself to 
be quite alone in the universe so that there is nothing to 
compare yourself with and then try to tell me how large 


you are. You cannot. You have no size unless something 
else exists for you to be larger or smaller than. 

So in any statement of physics we always have two objects 
in mind, die object we are primarily interested in and the 
object we are comparing it with. To simplify things we 
generally keep as far as possible to the same comparison 
object. Thus when we speak of size the comparison object is 
generally the standard metre or yard. Since we habitually 
use the same standard we tend to forget about it and scarcely 
notice that a second object is involved. We talk about the 
properties of an electron when we really mean the properties 
of an electron and a yard-stick properties which refer to 
experiences in which the yard-stick was concerned just as 
much as the electron. If we remember the second object at 
all we forget that it is a physical object; for us it is not a yard- 
stick, but just a yard. 

Primarily we say yard rather than yard-stick because a great 
many equivalent substitutes for the yard-stick are possible. 
But we do not generally think of a yard as a general name for 
one of a large variety of physical objects or systems; we do 
not think of it as an object at all. I grant that another 
physical object may be an equivalent substitute for a yard- 
stick, but I do not grant that a de-materialised yard is an 
equivalent substitute for a yard-stick. When the quantum 
physicist employs a standard of length in his theory, he does 
not treat it as an object; if he did, he would according to 
the principles of his theory have to assign a wave function 
to it, as he does to the other objects concerned in the 
phenomena. In my view he is wrong. Either he is using 
the standard length as a substitute for the second body 
concerned in the observed relation of size, in which case he 
ought to attribute to it a wave function, so that he can bring 
it into his equations in the same way that the second body 
would have been brought in; or he is treating size as though 
it were not an observable relation between one physical 


object and another, and the lengths referred to in his formulae 
are not the lengths which we try to observe.* 

We have to recognise then that what are called the 
properties of an electron are the combined properties or 
relations of the electron and some other physical system 
which constitutes a comparison object. For an electron by 
itself has no properties. If it were absolutely alone, there 
would be nothing whatever to be said about it not even 
that it was an electron. And we must not be misled by the 
fact that in current quantum theory the comparison object 
is replaced by an abstraction, e.g. a metre, which does not 
enter into the equations in the way that an observable com- 
parison object would do ; for that is a point on which current 
quantum theory is clearly at fault, "j" 

The progress of science depends on analysing our ex- 
perience into its simplest elements. An object of familiar 
experience such as a table is found to be highly complex, so 
we analyse it into molecules; the molecules are found to be 
complex and are analysed into atoms; the atoms are found 
to be complex and are analysed into protons and electrons. 
In the pursuit of simplification we reach smaller and smaller 
entities until, so far as we can tell, we arrive at the limit in 
the electron and proton. But what meanwhile is happening 
to the second object concerned in the experience the com- 
parison object ? Here our aim must be to substitute something 
more universal. We are dissatisfied with the yard-stick 
because it is clearly too local and specialised a system. To 
substitute an abstract yard is, as we have seen, a false step. 

* I do not mean that they have not the same numerical value; the 
quantum physicist secures that by empirical adjustment. But they are 
quantities of different nature, and the point of practical importance is 
that they have a different type of probability distribution. 

+ The ordinary current theory is not relativistic and does not profess 
to be the final form. The point, however, seems to have generally been 
overlooked by those who are attempting to formulate a fully relativistic 
quantum theory. We shall refer to it again, p. 245. 

ENPS 15 


We must continue to use a physical system for comparison 
idealised, if you like, but not to the extent of having a different 
relationship to human experience from that which a physical 
object has. We may use a system in which a yard (or any 
definite number of yards) figures as a characteristic, but not 
a disembodied yard. In the search for universality we pass 
from the earth to the sun, to the "mean of the stars", to the 
galaxy, and finally to the most universal of all systems, namely 
the universe itself. In this last system a definite number of 
yards figures as a characteristic which is called the radius of 
curvature of space-time; it is thus able to serve as a com- 
parison object for size. 

The end of our pursuit of simplicity is to reach as primary 
object the electron (or proton), and as comparison object the 
universe. It is to this combination that the simplest assertions 
refer, and the fundamental equations of physics in their 
simplest form apply. 

In present-day physics the most fundamental equation is 
the wave equation of an electron. It is usually supposed to 
describe the electron alone; but we have seen that that would 
be nonsense there is nothing to describe. It describes the 
relation of the electron to a physical comparison object or 
system; and although the comparison system is not men- 
tioned, we can easily see that it must be the universe not 
quite the actual universe, but the universe idealised by 
smoothing out all gravitational and electromagnetic fields. 
For if a more local comparison object were involved, wave 
mechanics would by its own principles employ a more 
complicated equation with a double wave function to exhibit 
the observable relations involving the electron and that 

Since the equation refers to conditions in which there is 
no gravitational field, the implied comparison universe is 
equally undisturbed. It must be remembered, however, that 
the wave equation has been found empirically from observa- 


tions made in the actual universe; the comparison object is 
not just any universe, containing as much or as little matter 
as we like to imagine. The smoothing out of gravitational 
fields is just the same idealisation as is used in Lemaitre's 
model of the expanding spherical universe; the stars and 
galaxies are smoothed out into a uniform distribution of 
matter; but the general dimensions are not tampered with. 
We may say briefly that in the wave equation the electron 
is referred to the Lemaitre spherical universe as comparison 

Thus the "wave equation of the electron" is an equation 
which straddles the whole of physics and describes the relation 
of the electron to the universe. If we invert the relation of 
the electron to the universe, we obtain the relation of the 
universe to the electron. We have only to take this equation 
describing the electron with the universe as comparison 
object, and view it, as it were, through the wrong end of the 
telescope, to obtain the equation describing the universe with 
the electron as comparison object. In describing the behaviour 
in particular, the expansion of the universe, the electron 
has virtually been our comparison standard; for the ordinary 
small-scale standards of length are constantly related to the 
electron. So in this way we arrive at an equation for the 
behaviour of the universe which (if the whole scheme of 
physics is consistent) must be equivalent to that given by 
relativity theory as developed by Friedman and Lemaitre; 
but instead of involving a cosmical constant of undetermined 
value, all its coefficients are definitely known; for they are 
taken from the wave equation of the electron of which it is 
another aspect. By comparison we can accordingly find the 
value of the cosmical constant. 

The procedure is not so simple as it sounds; but the 
difficulty is mainly that, before it can be applied, it is necessary 
to remove the fault (to which I have referred) in the existing 
quantum theory. Thus, although it is fairly simple in itself, 



it appears as the last step in a rather difficult investigation 
most of which is not directly concerned with the cosmical 

Work on these lines has convinced me that the subject of 
the expanding universe is not just an interesting side-track, 
but is on the main route of the future development of physics. 
It will have a practical importance in astronomy also; for 
if the value of the nebular recession calculated from the 
ordinary laboratory constants agrees with that found by 
astronomical observation, it will check the accepted scale 
of distances of the nebulae, which is at present somewhat 
doubtful. I do not wish to gloss over the fragmentary state 
of our present knowledge; but the subject of the expanding 
universe seems to me to deserve prominence as one that it 
is of the utmost importance to continue investigating. 


There has been a great deal of speculation in traditional philosophy which 
might have been avoided if the importance of structure, and the difficulty 
of getting behind it, had been realised. For example, it is often said that 
space and time are subjective, but they have objective counterparts; or 
that phenomena are subjective, but are caused by things in themselves, 
which must have differences inter se corresponding with the differences 
in the phenomena to which they give rise. Where such hypotheses are 
made, it is generally supposed that we can know very little about the 
objective counterparts. In actual fact, however, if the hypotheses , as 
stated were correct, the objective counterparts would form a world 

having the same structure as the phenomenal world In short, every 

proposition having a communicable significance must be true of both 
worlds or of neither: the only difference must lie in just that essence of 
individuality which always eludes words and baffles description, but 
which, for that very reason, is irrelevant to science. 

BERTH AND RUSSELL, Introduction to Mathematical Philosophy, p. 61. 


LET us suppose that a thousand years hence archaeologists 
are digging over the sites of the forgotten civilisation of 
Great Britain. They have come across the following literary 
fragment, which somehow escaped destruction when the 
abolition of libraries was decreed 

'Twas brillig, and the slithy toves 
Did gyre and gimble in the wabe, 

All mimsy were the borogoves 
And the mome raths outgrabe. 

This is acclaimed as an important addition to the scanty 
remains of an interesting historical period. But even the 
experts are not sure what it means. It has been ascertained 
that the author was an Oxford mathematician; but that does 


not seem wholly to account for its obscurity. It is certainly 
descriptive of some kind of activity; but what the actors are, 
and what kind of actions they are performing, remain an 
inscrutable mystery. It would therefore seem a plausible 
suggestion that Mr Dodgson was expounding a theory of 
the physical universe. 

Support for this explanation might be found in a further 
fragment of the same poem 

One, two! One, two! and through and through 
The vorpal blade went snicker-snack! 

"One, two! One, two!" Out of the unknown activities of 
unknown agents mathematical numbers emerge. The pro- 
cesses of the external world cannot be described in terms of 
familiar images; whether we describe them by words or by 
symbols their intrinsic nature remains unknown. But they 
are the vehicle of a scheme of relationship which can be 
described by numbers, and so give rise to those numerical 
measures (pointer-readings) which are the data from which 
all knowledge of the external universe is inferred. 

Our account of the external world (when purged of the 
inventions of the story teller in consciousness) must necessarily 
be a "J a bberwocky" of unknowable actors executing un- 
knowable actions. How in these conditions can we arrive at 
any knowledge at all? We must seek a knowledge which is 
neither of actors nor of actions, but of which the actors and 
actions are a vehicle. The knowledge we can acquire is 
knowledge of a structure or pattern contained in the actions. 
[ think that the artist may partly understand what I mean. 
(Perhaps that is the explanation of the Jabberwockies that 
we see hung on the walls of Art exhibitions.) In mathe- 
matics we describe such knowledge as knowledge of group 

It does not trouble die mathematician that he has to deal 
with unknown things. At the outset in algebra he handles 


unknown quantities x and y. His quantities are unknown, 
but he subjects them to known operations addition, multi- 
plication, etc. Recalling Bertrand Russell's famous definition, 
the mathematician never knows what he is talking about, 
nor whether what he is saying is true; but, we are tempted 
to add, at least he does know what he is doing. The last 
limitation would almost seem to disqualify him for treating 
a universe which is the theatre of unknowable actions and 
operations. We need a super-mathematics in which the 
operations are as unknown as the quantities they operate on, 
and a super-mathematician who does not know what he is 
doing when he performs these operations. Such a super- 
mathematics is the Theory of Groups. 

The Theory of Groups is usually associated with the 
strictest logical treatment. I doubt whether anyone hitherto 
has committed the sacrilege of wrenching it away from a 
setting of pure mathematical rigour. But it is now becoming 
urgently necessary that it should be tempered to the under- 
standing of a physicist, for the general conceptions and 
results are beginning to play a big part in the progress of 
quantum theory. Various mathematical tools have been tried 
for digging down to the basis of physics, and at present this 
tool seems more powerful than any other. So with rough 
argument and make-shift illustration I am going to profane 
the temple of rigour. 

My aim, however, must be very limited. At the one end 
we have the phenomena of observation which are somehow 
conveyed to man's consciousness via the nerves in his body; 
at the other end we have the basal entities of physics 
electrons, protons, waves, etc. which are believed to be the 
root of these phenomena. In between we have theoretical 
physics, now almost wholly mathematical. In so far as 
physical theory is complete it claims to show that the 
properties assigned to, and thereby virtually defining, the 
basal entities are such as to lead inevitably to the laws which 

ENPS *7 


we see obeyed in the phenomena accessible to our senses. 
If further the properties are no more than will suffice for this 
purpose and are stated in the most non-committal form 
possible, we may take the converse point of view and say 
that theoretical physics has analysed the universe of obser- 
vable phenomena into these basal entities. The working out 
of this connection is the province of the mathematician, and 
it is not our business to discuss it here. What I shall try to 
show is how mathematics first gets a grip on the basal entities 
whose nature and activities are essentially unknowable. We 
are to consider where the material for the mathematician 
comes from, and not to any serious extent how he mani- 
pulates the material. 

This limitation may unfortunately give to the subject an 
appearance of triviality. We express mathematically ideas 
which, so far as we develop them, might just as well have 
been expressed non-mathematically. But that is the only 
way to begin. We want to see where the mathematics jumps 
off As soon as the mathematics gets into its stride, it leaves 
the non-technical author and reader panting behind. I shall 
not be altogether apologetic if the reader begins to pant a 
little towards the end of the chapter. It is my task to show 
how a means of progress which begins with trivialities can 
work up momentum sufficient for it to become the engine 
of the expert. So in the last glimpse we shall have of it, we 
see it fast disappearing into the wilds. 


In describing the behaviour of an atom reference is often 
made to the jump of an electron from one orbit to another. 
We have pictured the atom as consisting of a heavy central 
nucleus together with a number of light and nimble electrons 
circulating round it like the planets round the sun. In the 
solar system any change of the orbit of a planet takes place 


gradually, but in the atom the electron can only change its 
orbit by a jump. Such jumps from one orbit to an entirely 
new orbit occur when an atom absorbs or emits a quantum 
of radiation (p. 37). 

You must not take this picture too literally. The orbits can 
scarcely refer to an actual motion in space, for it is generally 
admitted that the ordinary conception of space breaks down 
in the interior of an atom; nor is there any desire nowadays 
to stress the suddenness or discontinuity conveyed by the 
word "jump". It is found also that the electron cannot be 
localised in the way implied by the picture. In short, the 
physicist draws up an elaborate plan of the atom and then 
proceeds critically to erase each detail in turn. What is left 
is the atom of modern physics ! 

I want to explain that if the erasure is carefully carried out, 
our conception of the atom need not become entirely blank. 
There is not enough left to form a picture; but something is 
left for the mathematician to work on. In explaining how 
this happens, I shall take some liberties by way of simplifi- 
cation; but if I can show you the process in a system having 
some distant resemblance to an actual atom, we may leave it 
to the mathematician to adapt the method to the more 
complex conditions of Nature. 

For defmiteness, let us suppose that there are nine main 
roads in the atom nine possible orbits for the electron. Then 
on any occasion there are nine courses open to the electron; 
it may jump to any of the other eight orbits, or it may stay 
where it is. That reminds us of another well-known jumper 
the knight in chess. He has eight possible squares to move 
to, or he may stay where he is. Instead of picturing the atom 
as containing a particle and nine roads or orbits, why should 
we not picture it as containing a knight and a chess-board? 
"You surely do not mean that literally ! " Of course not ; but 
neither does the physicist mean the particle and the orbits to 
be taken literally. If the picture is going to be rubbed out, 



is it so very important that it should be drawn one way 
rather than another? 

It turns out that my suggestion would not do at all. 
However metaphorical our usual picture may be, it contains 
an essential truth about the behaviour of the atom which 
would not be preserved in die knight-chess-board picture. 
We have to formulate this characteristic in an abstract or 
mathematical way, so that when we rub out the false picture 
we may still have that characteristic the something which 
made the orbit picture not so utterly wrong as the knight 
picture to hand over to the mathematician. The distinction 
is this. If the electron makes two orbit jumps in succession 
it arrives at a state which it could have reached by a single 
jump; but if a knight makes two moves it arrives at a square 
which it could not have reached by a single move. 

Now let us try to describe this difference in a regular 
symbolic way. We must first invent a notation for describing 
the different orbit jumps. The simplest way is to number the 
orbits from i to 9, and to imagine the numbers placed con- 
secutively round a circle so that after 9 we come to I again. 
Then the jump from orbit 2 to orbit 5 will be described as 
moving on 3 places, and from orbit 7 to orbit 2 as moving 
on 4 places. We shall call the jump or operation of moving 
on one place Pi , of moving on two places P 2 , and so on. 
We shall then have nine different operators P, including the 
stay-as-you-were or identical operator P . 

We shall use the symbol A to denote the atom in some 
initial state, which we need not specify. Suppose that it 
undergoes the jump P* . Then we shall call the atom in the 
new state P*A\ that is to say, the atom in the new state is 
the result of performing the operation P on the system 
described as A. If the atom makes another jump P 4 , the 
atom in the resulting state will be described as P 4 P 2 ^4, since 
that denotes the result of the operation P 4 on the system 
described as Pa A. If we do not want to mention the particular 


( umps, but to describe an atom which has made two jumps 
from the original state A, we shall call it correspondingly 
P b P tt A; a and b stand for two of the numbers o, I, 2, . . . 8, 
but we do not disclose which. 

We have seen that two orbit jumps in succession give a 
state which could have been reached by a single jump. If 
the state had been reached by a single jump we should have 
called the atom in that state P C A, where c is one of the 
numbers o, i, 2, ... 8. Thus we obtain a characteristic pro- 
perty of orbit jumps, viz. they are such that 

P b P a A=P c A. 

Since it does not matter what was the initial state of the 
atom, and we do not pretend to know more about the atom 
than that it is the theatre of the operations P, we will divide 
the equation through by A, leaving 


This division by A may be regarded as the mathematical 
equivalent of the rubbing out of the picture. 

To treat the knight's moves similarly we may first dis- 
tinguish them as directed approximately towards the points 
of the compass N.N.E., E.N.E., E.S.E., and so on, and 
denote them in this order by the operators Qi , Q* , ... Qs . 
Qo will denote stay-as-you-were. Then since two knight's 
moves are never equivalent to one knight's move, our result 
will be* 

Q& Qa ^ Qc (unless a, b or c=o). 

We have to exclude c=o, because two moves might bring 
the knight back where it was originally. 

Let us spend a few moments contemplating this first result 
of our activities as super-mathematicians. The P's represent 
activities of an unknown kind occurring in an entity (called 
an atom) of unknown nature. It is true that we started with 

* The sign ^ means "is not equal to'* 


a definite picture of the atom with electrons jumping from 
orbit to orbit and showed that the equation P P 6 =P C was 
true of it. But now we have erased the picture; A has dis- 
appeared from the formula. Without the picture, the 
operations P which we preserve are of entirely unknown 
nature. An ordinary mathematician would want to be doing 
something definite to multiply, take square roots, differ- 
entiate, and so on. He wants a picture with numbers in it 
so that he can say for example that the electron has jumped 
to an orbit of double or n times the former radius. But we 
super-mathematicians have no idea what we are doing to the 
atom when we put the symbol P before A. We do not know 
whether we are extending it, or rotating it, or beautifying it. 
Nevertheless we have been able to express some truth or 
hypothesis about the activities of the atom by our equation 
P & P a P c . That our equation is not merely a truism is shown 
by the fact that when we start with a knight moving on a 
chess-board and make similar erasures we obtain just the 
opposite result Q 6 Q a ^ Q c . 

It happens that the property expressed by P b P a =P c is the 
one which has given the name to the Theory of Groups. 
A set of operators such that the product of any two of them 
always gives an operator belonging to the set is called a 
Group. Knight's moves do not form a Group. I am not 
going to lead you into the ramifications of the mathematical 
analysis of groups and subgroups. It is sufficient to say that 
what physics ultimately finds in the atom, or indeed in any 
other entity studied by physical methods, is the structure oj 
a set of operations. We can describe a structure without 
specifying the materials used; thus the operations that 
compose the structure can remain unknown. Individually 
each operation might be anything; it is the way they inter- 
lock that concerns us. The equation P 6 P a =P c is an example 
of a very simple kind of interlocking. 
The mode of interlocking of the operations, not their 


nature, is responsible for those manifestations of the external 
universe which ultimately reach our senses. According to 
our present outlook this is the basal principle in the philo- 
sophy of science. 

I must not mislead you into thinking that physics can 
derive no more than this one equation out of the atom, or 
indeed that this is one of the most important equations. But 
whatever is derived in the actual (highly difficult) study of 
the atom is knowledge of the same type, i.e. knowledge of 
the structure of a set of unknown operators. 


A very useful kind of operator is the selective operator. In my 
schooldays a foolish riddle was current "How do you catch 
lions in the desert?" Answer: "In the desert you have lots 
of sand and a few lions; so you take a sieve and sieve out the 
sand, and the lions remain". I recall it because it describes 
one of the most usual methods used in quantum theory for 
obtaining anything that we wish to study. 

Let Z denote the zoo, and Si the operation of sieving out 
or selecting lions; then S Z Z=L, where L denotes lions or, 
as we might more formally say, L denotes a pure ensemble 
having the leonine characteristic. These pure selective 
operators have a rather curious mathematical property, viz. 

S,'=S, (A). 

For Si 2 (an abbreviation for S^Sj) indicates that having 
selected all the lions, you repeat the operation, selecting all 
the lions from what you have obtained. Putting through 
the sieve a second time makes no difference; and in fact, 
repeating it n times, you have 5j n ==Sj. The property ex- 
pressed by equation (A) is called idempotency. 
Now let S t be the operation of selecting tigers. We have 

S,S,=o (B). 


For if you have first selected all the lions, and go on to select 
from these all the tigers, you obtain nothing. 

Now suppose that the different kinds of animals in the zoo 
are numbered in a catalogue from i to n and we introduce 
a selective operator for each; then 

Si+S*+S 3 +S 4 + . . . +S n =I (C), 

where I is the stay-as-you-were operator. For if you sieve 
out each constituent in turn and add together the results, 
you get the mixture you started with. 

A set of operators which satisfies (A), (B) and (C) is called 
a spectral set, because it analyses any aggregation into pure 
constituents in the same way that light is analysed by a 
prism or grating into the different pure colours which form 
the spectrum. The three equations respectively secure that 
the operators of a spectral set are idempotent, non-over- 
lapping and exhaustive. 

Let us compare the foregoing method of obtaining lions 
from the zoo with the method by which "heavy water " is 
obtained from ordinary water. In the decomposition of 
water into oxygen and hydrogen by electrolysis, the heavy 
water for some reason decomposes rather more slowly than 
the ordinary water. Consequently if we submit a large 
quantity of water to electrolysis, so that the greater part 
disappears into gas, the residue contains a comparatively high 
proportion of heavy water. This process of "fractionating" 
is a selective operation, but it is not pure selection such as 
we have been considering. If taking the residue we again 
perform the operation of electrolysis we shall still further 
concentrate the heavy water. A fractionating operator F is 
not idempotent (FVJF), and this distinguishes it from a pure 
selective operator S. 

The idea of analysing things into pure constituents and of 
distinguishing mixtures from pure ensembles evidently plays 
an important part in physical conceptions of reality. But it 


is not very easy to define just what we mean by it. We think 
of a pure ensemble as consisting of a number of individuals 
all exactly alike. But the lions at the zoo are not exactly 
alike; they are only alike from a certain point of view. Are 
the molecules of heavy water all exactly alike? We cannot 
speak of their intrinsic nature, because of that we know 
nothing. It is their relations to, or interactions with, other 
objects which define their physical properties; and in an 
interrelated universe no two tilings can be exactly alike in 
all their relations. We can only say then that the molecules 
of heavy water are alike in some common characteristic. But 
that is not sufficient to secure that they form a pure en- 
semble; the molecules which form any kind of mixture are 
alike in one common characteristic, viz. that they are mole- 

If we have a difficulty in defining purity of things for which 
we have more or less concrete pictures, we find still more 
difficulty with regard to the more recondite quantities of 
physics. Nevertheless it is clear that the idea of distinguishing 
pure constituents from mixtures contains a germ of important 
truth. It is the duty of the mathematician to save that germ 
out of the dissolving picture; and he does this by directing 
attention not to the nature of what we get by the operation 
but to the nature of the selective operation itself. He shows 
that those observational effects which reach our perceptions, 
generally attributed to the fact that we are dealing with an 
assembly of like individuals, are deducible more directly 
from the fact that the assembly is obtainable by a kind of 
operation which, once performed, can be repeated any 
number of times without making any difference. He thus 
substitutes a perfectly definite mathematical property of the 
operator, viz. SfSi, for a very vaguely defined property 
of the result of the operation, viz. a certain kind of likeness 
of the individuals which together form L. He thus frees his 
results from various unwarranted hypotheses that may have 


been introduced in trying to form a picture of this property 

In the early days of atomic theory, the atom was defined 
as an indivisible particle of matter. Nowadays dividing the 
atom seems to be the main occupation of physicists. The 
definition contained an essential truth; only it was wrongly 
expressed. What was really meant was a property typically 
manifested by indivisible particles but not necessarily con- 
fined to indivisible particles. That is the way with all models 
and pictures and familiar descriptions ; they show the property 
that we are interested in, but they connect it with irrelevant 
properties which may be erroneous and for which at any rate 
we have no warrant. You will see that the mathematical 
method here discussed is much more economical of hypo- 
thesis. It says no more about the system than that which it 
is actually going to embody in the formulae which yield the 
comparison of theoretical physics with observation. And, 
in so far as it can surmount the difficulties of investigation, 
its assertions about the physical universe are the exact 
systematised equivalent of the observational results on which 
they are based. I think it may be said that hypotheses in the 
older sense are banished from those parts of physical science 
to which the group method has been extended. The modern 
physicist makes mistakes, but he does not make hypo- 

One effect of introducing selective operators is that it 
removes the distinction between operators and operands. In 
considering the "jump" operators P, we had to introduce 
an operand A, for them to work on. We must furnish some 
description of A, and A is then whatever answers to that 
description. Let S a be the operation of selecting whatever 
answers to the description A, and let U be the universe. 
Then evidently A= S a U\ and instead of P 6 P a A we can write 
P b P a S a U. Thus special operands, as distinct from operators, 
are not required. We have a large variety of operators, some 


of them selective, and just one operand the same in every 
formula namely the universe. 

This mathematical way of describing everything with 
which we deal emphasises, perhaps inadvertently, an im- 
portant physical truth. Usually when we wish to consider a 
problem about a hydrogen atom, we take a blank sheet of 
paper and mark in first the proton and then the electron. That 
is all there is in the problem unless or until we draw something 
else that we suppose to be present. The atom thus presents 
itself as a work of creation a creation which can be stopped 
at any stage. When we have created our hydrogen atom, we 
may or may not go on to create a universe for it to be part of. 
But the real hydrogen atoms that we experiment on are 
something selected from an always present universe, often 
selected or segregated experimentally, and in any case 
selected in our thoughts. And we are learning to recognise 
that a hydrogen atom would not be what it is, were it not 
the result of a selective operation performed on that maze of 
interrelatedness which we call the universe. 

In Einstein's theory of relativity the observer is a man who 
sets out in quest of truth armed with a measuring-rod. In 
quantum theory he sets out armed with a sieve. 


I am now going to introduce a set of operations with which 
we can accomplish something rather more ambitious. They 
are performed on a set of four things which I will represent 
by the letters A, B, C, D. We begin with eight operations; 
after naming (symbolically) and describing each operation, 
I give the result of applying it to ABCD: 

5 a . Interchange the first and second, also the third and 
fourth. BADC. 

S0. Interchange the first and third, also the second and 
fourth. CDAB. 


S y . Interchange the first and fourth, also the second and 
third. DCBA. 

5 . Stay as you were. ABCD. 
D a . Turn the third and fourth upside down. ABDG. 
Dp. Turn the second and fourth upside down. AffCQ. 
Dy . Turn the second and third upside down. A9DD. 
Dg . Stay as you were. ABCD. 

We also use an operator denoted by the sign which means 
"turn them all upside down". 

We can apply two or more of these operations in suc- 
cession. For example, S a Sp means that, having applied the 
operation Sp which gives CDAB, we perform on the result 
the further operation S a which interchanges the first and 
second and also the third and fourth. The result is DCBA. 
This is the same as the result of the single operation 5 y ; 

S a Sj8=S y . 

Sometimes, but not always, it makes a difference which of 
the two operations is performed first. For example, 

Taking the result of the operation D y , viz. AffDD, and 
performing on it the operation 5 a , we obtain ff ADD. 

But taking the result of the operation S a , viz. BADC, and 
performing on it the operation D y , we obtain BVQC. 

Thus the double operation S a D y is not the same as D y 5 a . 
There is, however, a simple relation, ff ADD is obtained by 
inverting each letter in BVQC, that is to say, by applying 
the operation which we denote by the sign . Thus 

Operators related in this way are said to anticommute. On 
examination we find that 5 a , Sp commute, and so do D a , Dp; 
so also do 5 a and D a . It is only a combination of an S and 
a D with different suffixes a, ft, y (but not 8) which exhibits 


We can make up sixteen different operators of the form 
S a A, , where a and b stand for any of the four suffixes a, /?, y , 
8. It is these combined operators which chiefly interest us. 
I will call them ^-operators and denote them by JEi, Ea, 
3,. . .16. They form a Group, which (as we have seen) 
means that the result of applying two operations of the Group 
in succession can equally be obtained by applying a single 
operation of the Group. I should, however, mention that 
the operation is here regarded as thrown in gratuitously.* 
We may not by a single operation E c be able to get the 
letters into the same arrangement as that given by E^ E a ; but 
if not, we can get the same arrangement with all the letters 
inverted. This property of the -operators is accordingly 
expressed by 

JL / 71 T-I . -f-t 

E a E b =E c . 

We now pick out five of the E-operators. Our selection 
at first sight seems a strange one, because it has no apparent 
connection with their constitution out of S- and D- operators. 
It is as follows 

I== S a D a , which gives BADC. 

Ei=S y D y DDSA. 

E 4 =SD y 3 ADD. 

E s =S y Dp (3CSA. 

These five are selected because they all anticommute with 
one another; that is to say, Ei Ei = Ez E\ , and so on for all 
the ten pairs. You can verify this by operating with the four 
letters, though, of course, there are mathematical dodges for 
verifying it more quickly. We call a set like this a pentad. 

There are six different ways of choosing our pentad, 
obtained by ringing the changes on the suffixes a, j8, y. But 
it is not possible to find more than five E-operators each of 

* To obtain a Group according to the strict definition we should have 
to take 32 operators, viz, die above 16, and the 16 obtained by prefixing - . 


which anticommutes with all the others. That is why we 
have to stop at pentads. 

Another important property must be noticed. You will 
see at once that i a =i; for Ei is the same as S a , and a 
repetition of the interchange expressed by 5 a restores the 
original arrangement. But consider Ef. In the operation 5 
we turn the second and fourth letters upside down, and 
then reverse the order of the letters. The letters left right 
way up are thereby brought into the second and fourth 
places, so that in repeating the operation they become 
turned upside down. Hence the letters come back to their 
original order, but are all upside down. This is equivalent 
to the operation . So that we have JE 5 2 = i. In this way 
we find that 

J7 2 172 172 T 172 172 T 

JCl =}2 ~ 3 I, 24 JC5 I. 

A pentad always consists of three operators whose square 
is i, and two operators whose square is i. 

With regard to the symbols i and i, I should explain 
that i here stands for the stay-as-you-were operator. Since 
that is the effect of the number i when it is used as an 
operator (a multiplier) in arithmetic, the notation is appro- 
priate. (We have also denoted the stay-as-you-were operator 
by 5g and Dg, so that we have S$=D$ = i.) Since the 
operator i makes no difference, the operators " " and 
" i " are the same; so we sometimes put in a i, when by 
itself would look lonely. Repetition of the operation 
restores the original state of things; consequently ( ) z or 
( i)* is equal to i. Although the symbol, as we have here 
defined it, has no connection with "minus", it has in this 
respect the same property as and i in algebra. 

I have told you that the proper super-mathematician never 
knows what he is doing. We, who have been working on a 
lower plane, know what we have been doing. We have been 
Ipusy rearranging four letters. But there is a super-mathe- 


matician within us who knows nothing about this aspect of 
what we have been studying. When we announce that we 
have found a group of sixteen operations, certain pairs of 
which commute and the remaining pairs anticommute, some 
of which are square roots of I and the others square roots 
of i, he begins to sit up and take notice. For he can grasp 
this kind of structure of a group of operations, not referring 
to the nature of the operations but to the way they interlock. 
He is interested in the arrangement of the operators to form 
six pentads. That is his ideal of knowledge of a set of 
operations knowledge of its distinctive kind of structure. 
A great many other properties of ^-operators have been 
found, which I have not space to examine in detail. There 
are pairs of triads, such that members of the same triad all 
anticommute but each commutes with the three members 
of the opposite triad. There are anti-triads composed of three 
mutually commuting operators, which become anti-tetrads 
if we include the stay-as-you-were operator. 

All this knowledge of structure can be expressed without 
specifying the nature of the operations. And it is through 
recognition of a structure of this kind that we can have 
knowledge of an external world which from an ordinary 
standpoint is essentially unknowable. 

Some years ago I worked out the structure of this group 
of operators in connection with Dirac's theory of the electron. 
I afterwards learned that a great deal of what I had written 
was to be found in a treatise on Rummer's quartic surface. 
There happens to be a model of Kummer's quartic surface 
in my lecture-room, at which I had sometimes glanced with 
curiosity, wondering what it was all about. The last thing 
that entered my head was that I had written (somewhat 
belatedly) a paper on its structure. Perhaps the author of the 
treatise would have been equally surprised to learn that he was 
dealing with the behaviour of an electron. But then, you see, 
we super-mathematicians never do know what we are doing. 


As the result of a game with four letters we have been able 
to describe a scheme of structure, which can be detached from 
the game and given other applications. When thus detached, 
we find this same structure occurring in the world of physics. 
One small part of the scheme shows itself in a quite elementary 
way, as we shall presently see; another part of it was brought 
to light by Dirac in his theory of the electron; by further 
search the whole structure is found, each part having its 
appropriate share in physical phenomena. 

When we seek a new application for our symbolic 
operators , we cannot foresee what kind of operations they 
will represent; they have been identified in the game, but 
they have to be identified afresh in the physical world. Even 
when we have identified them in the familiar story of con- 
sciousness, their ultimate nature remains unknown; for the 
nature of the activity of the external world is beyond our 
apprehension. Thus armed with our detached scheme of 
structure we approach the physical world with an open mind 
as to how its operations will manifest themselves in our 

I shall have to refer to an elementary mathematical result. 
Consider the square of (2JEi + S-E*), that is to say the operation 
which is equivalent to twice performing the operation 
(2^1+ 3^2 ). We have not previously mixed numbers with 
our operators; but no difficulty arises if we understand that 
in an expression of this kind 2 stands for the operation of 
multiplying by 2, 3 for the operation of multiplying by 3, as 
in ordinary algebra. We have 

We have had to attend to a point which does not arise in 
ordinary algebra. In algebra we should have lumped together 
the two middle terms and have written i2EiE 2 instead of 


6Ei E 2 +6E 2 Ei . But we have seen (p. 269) that the operation 
E 2 followed by the operation Ei is not the same as the 
operation Ei followed by the operation E 2 ; in fact we de- 
liberately chose these operators so that E 2 Ei = EiE 2 . 
Consequently the two middle terms cancel one another and 
we are left with 

But we have also seen that JBi 2 =i, E 2 2 =i. Thus 

In other words (2Ei + ^E 2 ) is the square root of 13, or rather 
a square root of 13. 

Suppose that you move to a position 2 yards to the right 
and 3 yards forward. By the theorem of Pythagoras your 
resultant displacement is (/ (2? + 3 2 ) or V 1 3 yards. It suggests 
itself that when the super-mathematician (not knowing what 
kind of operations he is referring to) says that (2Ei + $E 2 ) is 
a square root of 13, he may mean the same thing as the 
geometer who says that a displacement 2 yards to the right 
and 3 yards forward is square~root-o-i3 yards. Actually 
the geometer does not know what kind of operations he is 
referring to either; he only knows the familiar story teller's 
description of them. He can render himself independent of 
the imaginations of the familiar story teller by becoming a 
super-mathematician. He will then say: 

What the familiar story teller calls displacement to the 
right is an operation whose intrinsic nature is unknown to 
me and I will denote it by Ei ; what he calls displacement 
forward is another unknown operation which I will denote 
by E^ . The kind of knowledge of the properties of displace- 
ment which I have acquired by experience is contained in 
such statements as "a displacement 2 yards to the right and 
3 yards forward is square-root-of-13 yards". In my notation 
this becomes "2Ei + $E 2 is a square-root of 13". Super- 
mathematics enables me to boil down these statements to 
ENPS 18 


the single conclusion that displacement to the right and dis- 
placement forward are two operations of the set whose group 
structure has been investigated in Section iv.* 

Similarly we can represent a displacement of 2 units to the 
right, 3 units forward and 4 units upward by (zEi + 3 2 + 4^3 ) . 
Working out the square of this expression in the same 
way, the result is found to be 29, which agrees with the 
geometrical calculation that the resultant displacement is 
V(2 2 +3 2 +4 2 )=V 2 9- The secret is that the super-mathe- 
matician expresses by the anticommutation of his operators the 
property which the geometer conceives as perpendicularity of 
displacements. That is why on p. 269 we singled out a pentad 
of anticommuting operators, foreseeing that they would 
have an immediate application in describing the property of 
perpendicular directions without using the traditional picture 
of space. They express the property of perpendicularity 
without the picture of perpendicularity. 

Thus far we have touched only the fringe of the structure 
of our set of sixteen E-operators. Only by entering deeply 
into the theory of electrons could I show the whole structure 
coming into evidence. But I will take you one small step 
farther. Suppose that you want to move 2 yards to the 
right, 3 yards forward, 4 yards upward, and 5 yards per- 
pendicular to all three in a fourth dimension. By this time 
you will no doubt have learned the trick, and will write 
down readily (2Ei + ^E 2 +4E 3 + sE^) as the operator which 
symbolises this displacement. But there is a break-down. 
The trouble is that we have exhausted the members of the 
pentad whose square is i, and have to fall back on E 4 whose 
square is I (p. 270). Consequently 

* He will, of course, require more than a knowledge relating to two 
of the operators to infer the group structure of the whole set. The 
immediate inference at this stage is such portion of the group structure 
as is revealed by the equations 1* = EJ = i , 2 1 = - j E* . 


Thus our displacement is a square root of 4, whereas Pytha- 
goras's theorem would require that it should be the square 
root of 2 2 +3 2 +4 2 +5 2 , or 54. Thus there is a limitation to 
our representation of perpendicular directions by B-operators ; 
it is only saved from failure in practice because in the actual 
world we have no occasion to consider a fourth perpen- 
dicular direction. How lucky ! 

It is not luck. The structure which we are here discussing 
is claimed to be the structure of the actual world and the key 
to its manifestations in our experience. The structure does 
not provide for a fourth dimension of space, so that there 
cannot be a fourth dimension in a world built in that way. 
Our experience confirms this as true of the actual universe. 

If we wish to introduce a fourth direction of displacement 
we shall have to put up with a minus sign instead of a plus 
sign, so that it will be a displacement of a somewhat different 
character. It was found by Minkowski in 1908 that "later" 
could be regarded in this way as a fourth direction of dis- 
placement, differing only from ordinary space displacements 
in the fact that its square combines with a minus instead of 
a plus sign. Thus 2 yards to the right, 3 yards forward, 4 yards 
upward and 5 "yards" later* is represented by the operator 
(2E I + 3jB 2 +4E3 + 5JB 4 ). We have calculated above that it is 
a square root of 4, so that it amounts to a displacement of 
2 yards. When, as here, we consider displacement in time 
as well as in space, the resultant amount is called the interval 
The value of the interval in the above problem according to 
Minkowski's formula is 2 yards, so that our results agree. 
Minkowski introduced the minus instead of the plus sign in 
the fourth term, regarding the change as expressing the 
mathematical distinction between time and space; we intro- 
duce it because we cannot help it it is forced on us by the 
group structure that we are studying. Minkowski's interval 

* A "yard" of time is to be interpreted as the time taken by light to 
travel a yard. 

1 8-2 


afterwards became the starting point of the general theory 
of relativity. 

Thus the distinction between space and time is already 
foretold in the structure of the set of E-operators. Space can 
have only three dimensions, because no more than three 
operators fulfil the necessary relationship of perpendicular 
displacement. A fourth displacement can be added, but it has 
a character essentially different from a space displacement. 
Calling it a time displacement, the properties of its associated 
operator secure that the relation of a time displacement to 
a space displacement shall be precisely that postulated in the 
theory of relativity. 

I do not suggest that the distinction between the fourth 
dimension and the other three is something that we might 
have predicted entirely by a priori reasoning. We had no 
reason to expect a priori that a scheme of structure which we 
found in a game with letters would have any importance in 
the physical universe. The agreement is only impressive if 
we have independent reason to believe that the world- 
structure is based on this particular group of operators. We 
must recall therefore that the E-operators were first found 
to be necessary to physics in Dirac's wave equation of an 
electron. Dirac's great achievement in introducing this 
structure was that he thereby made manifest a recondite 
property of the electron, observationally important, which 
is commonly known as its "spin". That is a problem which 
seems as far removed as possible from the origin of the 
distinction of space and time. We may say that although the 
distinction of space and time cannot be predicted for a 
universe of unknown nature, it can be predicted for a uni- 
verse whose elementary particles are of the character de- 
scribed in modern wave mechanics. 

It only remains to add that the sixteen E-operators are 
those referred to in the previous chapter (p. 236). We there 
use a double set; e.g. if Ei signifies the operation of displacing 


electron No. I to the right, Fi denotes the operation of 
displacing electron No. 2 to the right. I have already ex- 
plained the way in which the mystic number 136 arises in 
this double set. The double set of operators is not confined 
to the particular problem of two particles. It has universal 
application owing to the fact that we can only observe 
relations; therefore our standard equipment consists of two 
sets of operators, one for each end of the relation. For that 
reason the number 136 appears again in a different connection 
on p. 247. 

There are two ways in which the number 136 is involved 
in a double-set of E-operators. I have explained one way; 
I will now explain the other way, which is I believe more 
directly the origin of its occurrence in the fundamental 
constants of Nature. We have distinguished operators such 
as Ei whose square is i from those such as 4 whose square 
is i. We may call the first "space-like" and the second 
"time-like", since that is the way in which the distinction 
has appeared so far as we have investigated it in the foregoing 
discussion. Classifying all the 1 6 JS-operators in this way, 
we find that 10 of them are space-like and 6 are time- 
like. Classifying similarly the 256 EF-operators, we find 
at once that io 2 +6 2 or 136 are space-like (square i), and 
(10 x 6) + (6 x 10) or 120 are time-like (square = i). I think 
that when 136 occurs in the constants of physics it generally 
refers to the 136 space-like double operators; for the space- 
like operators determine the number of dimensions of the 
domain over which the total probability of a system is 
distributed. But to pursue these questions would take us too 
deeply into the theory. 


Codlin's the friend, not Short. Short's very well as far as he goes, but 
the real friend is Codlin not Short. 

DICKENS, The Old Curiosity Shop. 


IN the early days of die theory of relativity one of the most 
frequent questions asked by my correspondents was, Is the 
FitzGerald contraction real or apparent ? Is it really true that 
a rapidly moving rod becomes shortened in the direction of 
its motion? The answer which I gave in The Nature of the 
Physical World (pp. 32-34) is too long to quote here; but, 
having pointed out with an example that we often draw a 
distinction between things which are "true " and things which 
are "really true", I explained that on the same principles the 
contraction of the moving rod would be described as true 
but not really true. 

It is interesting to note the reaction of a not unfriendly 
philosophical reviewer* towards this effort to explain: 

. Surely it is simpler to say straight out that distance between 
two particles is not a dual relation but a triple relation into which 
a frame of reference enters, and that the only valuable dual 
relation that can be extracted is in cases when the two particles 
are relatively at rest and are permitted to fix the frame of reference 
as one Li which they are relatively at rest. 

Very much simpler for the author, at any rate. 

Non-technical books are very often a target for criticism 
simply because they are non-technical. I have quoted a very 
innocent example of such criticism. But one does not so 

* Mind, vol. 38, p. 413 (1929). 


easily excuse the critic who imputes scientific laxity on no 
better grounds. Statements are described as careless because 
they are not hedged about with safeguards like legal docu- 
ments. Explanations are treated as definitions. The author is 
convicted of not saying what he means. Of course he does 
not say exactly what he means; in ordinary speech one 
seldom does. The understanding between a non-technical 
writer and his reader is that he shall talk more or less like a 
human being and not like an Act of Parliament. 

I take it that the aim of such books must be to convey 
exact thought in inexact language. The author has abjured 
the technical terms and mathematical symbols which are the 
recognised means of securing exact expression, and he is 
thrown back on more indirect methods of awakening in the 
mind of the reader the thought which he wishes to convey. 
He will not always succeed. He can never succeed without 
the cooperation of the reader. 

A correspondent has pointed out to me that in various 
places in The Nature of the Physical World the word " space" 
occurs with four different meanings. I think he expected me 
to feel penitent. But the word has these meanings; and 
inasmuch as my correspondent was by no means diffident 
in telling me what I really meant in each place, I inferred 
that in this instance my attempt to convey exact thought in 
inexact language had succeeded. 

It is not a question of stepping down from the austere 
altitude of scientific contemplation to a plane of greater 
laxity. To free our results from pedantries of expression, and 
to obtain an insight in which the less essential complications 
do not obtrude, is as necessary in research as in public ex- 
position. We strive to reduce what we have ascertained to 
an exact formulation, but we do not leave it buried in its 
formal expression. We are continually drawing it out from 
its retreat to turn it over in our minds and make use of it 
for further progress; and it is in this handling of the truth 


that the rigour of scientific thought especially displays itself. 
Rigour is a much misused term, and not only in expository 
writing but in original scientific investigations it is too fre- 
quently another name for lack of a sense of proportion. 


In reading the various discussions which have arisen out of 
the philosophical position that I have taken up in my earlier 
writings, it has seemed to me that the most urgent point of 
controversy is the deadlock referred to in Chapter I con- 
cerning my remark "Mind is the first and most direct thing 
in our experience; all else is remote inference". The typical 
philosopher and the typical scientist seem to have taken up 
irreconcilable positions. 

First let me summarise my own view which is, I think, 
acceptable to most scientific men who have reflected at all 
on the subject. The experience of each individual is primarily 
the changing content of his consciousness. It is a succession 
of tableaux accompanied by sensory feelings of various kinds, 
memories, abstract thoughts, emotions, etc. Even to the 
least reflective of us this complex activity presents a problem; 
we want to find die associations of the various elements in 
this experience, and to make out what it is all about. But for 
the scientist at least the nature of the inquiry is very largely 
determined by the fact that individuals are able to com- 
municate part of their experience to one another. When you 
tell me your experience the sound of your voice is part of 
my experience; but for the purposes of the problem it is not 
my awareness of a sound that I utilise, but your awareness of 
something else. I treat it as an admission to an experience a 
content of consciousness which is not my own. By this 
step the problem is enlarged; it is no longer a matter of 
determining the interrelatedness of elements all contained in 
one consciousness, but the interrelatedness of elements in 


many different consciousnesses; it therefore requires the con- 
ception of a theatre of activity external to the individual 
consciousness. Physical science gives us a picture, or more 
strictly a symbolic formulation, of such an external theatre 
of activity interacting with each individual consciousness. 
If we accept the scientific solution, and in particular the 
scientific account of the nerve mechanism of the body, the 
connection between the objects inferred to exist in this 
external world and the sensations experienced in conscious- 
ness is evidently remote and indirect. 

It is very difficult to see what the philosopher is after when 
he challenges this. It is important to discover whether it is 
simply the common kind of misunderstanding which arises 
when two people do not talk the same language, or whether 
the philosopher really intends to reject the scientific account 
of the origin of our perceptions so far as that origin lies 
outside consciousness itself. 

Two philosophical writers have entered into this question in 
some detail with special reference to my own writings, namely 
Prof. W. T. Stace* and Mr C. E. M. Joad. f I will deal with 
them separately since they take up rather different positions. 
I understand that between them they represent a considerable 
body of opinion among philosophers of the realist school. 

I might define my typical opponent as the man who 
believes in the existence outside the mind of "an actual apple 
with an actual taste in it". I do not object to an actual apple 
external to the mind, and I am willing to be convinced as to 
the existence of an actual taste (as distinct from the physical 
interaction between the molecules of the apple and those of 
a particular tongue) external to the mind. Where the philo- 
sopher seems to fly against the plain teaching of science is in 
locating the actual taste in the actual apple. It is better to 
avoid words such as taste, colour, sound, which are used 

* " Sir Arthur Eddington and the Physical World", Philosophy, vol. 9, 
p. 40 (1934). f Philosophical Aspects of Modern Science (1932). 


confusingly both for the sensation and the indirect cause of 
the sensation. It might be clearer therefore if I described the 
philosophers in question as those who believe in u an actual 
dentist's drill with an actual pain in it", which is, I suppose, 
an obvious corollary. But I am afraid of saddling them with 
opinions which I have not seen explicitly admitted. 

The kind of datum from which scientist and philosopher 
alike must start is exemplified by I-perceivc-the-taste-of-an- 
apple. I use this string of words to indicate to you a particular 
kind of awareness ; it is the awareness, not the description 
nor the analysis implied in the description, which constitutes 
the datum. Another datum may be he-perceives-the-sound- 
of-a-bell. It is agreed that although I have become possessed 
of this second datum in an indirect way it ranks equally with 
the first which is immediately furnished by my consciousness. 
The recognition that sense-data may have different subjects 
("I" and "he'*) is the first step in their analysis. It suggests 
itself for trial that they may be treated as subject-object 
relations. We can form a kind of equation 

Datum minus Subject minus a constant relation ("per- 
ceives ") = Object. 

But if this is to carry us any further we must suppose that 
some at least of the objects are capable of association with 
different subjects. We then have a communal object the-sound- 
of-a-bell which is common to the data I-perceive-the-sound- 
of-a-bell and he-perceives-the-sound-of-a-bell; just as the 
subject "I" can be common to a number of data. The view 
that the objects of sense-data are communal objects, capable 
of perception by more than one subject, is a hypothesis; it 
implies that the gamut of sensations of one individual can in 
some way be identified with those of another individual. 

The data are evidently mental; they are an awareness a 
content of the consciousness of myself or of so-and-so. The 
communal objects, if they exist, are not in any one con- 


sciousness; nor are they to be identified with the objects of 
the physical world. If it is necessary to locate them any- 
where it must be in some third territory. Prof. Stace suggests 
the following as a view to which the physicist ought not to 

The view that sensory qualities are mental depends on the 
uncriticized dogma that there are only two realms to which they 
could belong, the physical and the mental. If this were so, then 
to prove that they are not physical would be the same as proving 
that they are mental. And this is what the physicist does. But 
the assumption on which the argument is based, the traditional 
common-sense division of the universe into mind and matter and 
nothing else is false. There is a third realm, which is neither 
physical nor mental, but which we may call the "neutral" realm. 
Sensory qualities belong to this realm and are neither physical 
nor mental. 

It is true that I commonly use the words mind and mental 
to cover all that is non-physical in the same way that matter 
and material arc frequently used to cover all physical entities. 
There is I think no more comprehensive term which would 
be generally intelligible. The paucity of language is illustrated 
by Prof. Stace's suggested term neutral. It suggests that the 
sensory qualities (i.e. communal objects) are neutral as be- 
tween the physical and the mental realm; and it is evident 
from later statements that that is how Stace regards them. 
But from a physicist's point of view they are ultra-mental. 
I mean that when from our common sense-data the philo- 
sopher derives a communal object the-sound-of-a-bell, and 
still more when he derives (I know not how) a bell with a 
sound in it, he is proceeding away from the external world 
of physics; to reach the bell contemplated by the physicist 
these steps have to be retraced and we have to start again 
from the immediate awareness of individual minds as I shall 
show later. Prof. Stace continues 

From this point of view Sir Arthur's statement, "Mind is the 


first and most direct thing in our experience; all else is remote 
inference'* which, he says, horrifiecf the philosophers so much 
would appear to have nothing horrifying in it except the 
apparent identification of sense-data with minds. That, I suppose, 
was what shocked the philosophers. 

I do not admit that the philosopher's construct of a realm 
of sensory qualities (communal objects) existing outside our 
individual minds is any less a remote inference than the 
construct of a physicist from the same data. An "impersonal 
taste" not forming part of the content of anyone's con- 
sciousness may be a legitimate conception, but it is not a 
matter of immediate experience; at least I have never tasted 
one. I must make it clear that for me sensory data are the 
experiences themselves the awareness of someone of some- 
thing; whereas Stace and perhaps philosophers generally use 
the term sense-data for what I here call communal objects 
such as the-taste-of-an-apple, or even for an object supposed 
to be compounded of tastes, colours, shapes, etc. like the 
familiar apple described by the story teller in my con- 
sciousness. To call such hypothetical constructs "data" seems 
to me to beg the question. 

Let us trace the first few steps in a scientist's inference from 
the data of his experience. In the first place he finds the data 
in which he himself plays the part of subject arranged in a 
time sequence. A more primitive description of a state of 
awareness would be I-perceive-a-taste-which-I-perceived- 
yesterday. We recognise a sensation not for what it is in 
itself but for its resemblance to a previous sensation. Attention 
at once passes away from the nature of the experience to the 
recurrency of the experience. As shown in the first chapter, 
it is out of the recurrencies of experience that the world of 
physics is inferred. The scientist's path therefore diverges 
immediately from that of the philosopher. The latter forms 
a general object or sensory quality, the-taste-of-an-apple, not 
in any one person's experience and therefore not associated 


with any particular place or time; any recWrrency is lost by 
removing it from the time sequence of its subject. It may 
happen that a number of individuals (bent on keeping the 
doctor away) perceive the-taste-of-an-apple with diurnal 
recurrency. It seems to me that the philosopher who starts 
by treating the-taste-of-an-apple as the object of an ex- 
perience common to them all and therefore dissociated from 
individual time sequences is unable to deal with this re- 
currency even though it is a recurrency of their common 

The point that I have to bring out is that the communal 
objects or sensory qualities have no bearing on physical 
science, since they eliminate the very part of experience with 
which the physicist is concerned, namely recurrency, and 
retain the part with which he is not concerned, namely the 
qualitative character of the sensations. Therefore the sensory 
qualities discussed by the philosophers do not form a realm 
intermediate between the mental and the physical world. 

At first sight it seems very satisfactory that the philosopher 
should care for that which the physicist neglects and vice 
versa. But I feel bound to ask the question whether the 
philosopher starting from the same data of experience has 
deliberately tackled a different problem and reached a 
different domain of truth, or whether he has attempted the 
same problem of synthesis as the physicist and has lost his 
way at the start. When the philosopher gives externality 
and objectivity to particular tastes I am not convinced that 
his basis lies in philosophy at all; it looks to me more like 
second-hand physics. 

The recognition of a taste of a particular quality as existing 
outside an individual consciousness must depend on identi- 
fying a sensation of taste by one individual with a sensation 
of taste by another individual. Therefore I take it that the 
philosopher supposes the taste sensations of different indi- 
viduals to have a one-to-on^ correspondence, and he in- 


terprets this correspondence as signifying identity of the 
object the taste perceived. Now the scientist also places 
the taste sensations of different individuals in one-to-one 
correspondence, without however speculating as to whether 
corresponding sensations are identical. For him corresponding 
(but not necessarily identical) sensations of taste are those 
which arise when the tongues of different people are stimu- 
lated by like objects. Is it this correspondence which the 
philosopher has taken over and reinterpreted? If not, how 
does he define his correspondence? Prof. Stace's view seems 
to be that there is floating round in his third " neutral" realm 
a particular taste capable of being perceived either by you 
or by me. If the taste is one that I perceived on a specified 
occasion, how are we to know when you perceive the same 
taste? The question demands an answer; for clearly the 
existence of a taste common to the perception of both of us 
cannot be one of the " first things in our experience" if we 
do not even know when the experience occurs. 

When a philosopher describes the taste as being the taste 
of an apple, it looks as though he had borrowed the scientist's 
criterion of correspondence. He apparently refers to the 
scientist's test of placing portions of an apple, i.e. a hard, 
round, green object with specified antecedents in the physical 
world, on the tongues of various individuals; and argues that 
because the physical conditions are similar the sensation in 
each mind is a perception of a common object of perception. 
But if that is really the way in which the philosopher dis- 
covers (or invents) the impersonal tastes which occupy his 
non-mental non-physical realm, these tastes, so far from 
being immediately known to exist, are a remote inference 
from our remote inferences about the physical world. 

If every element of experience was utterly unlike every 
other element of experience there would be no subject- 
matter either for science or philosophy. Progress depends 
on recognising a common factor in two elements of ex- 


perience. The most elementary type of common factor is 
indicated by our recognition that a number of experiences 
may have the same subject; we may pass over this, since no 
controversy arises. After this we must study further re- 
semblances between two elements of experience, either 
(a) which have the same subject, or (b) which have different 
subjects. Physics is based on (a) ; neo-realist philosophy seems 
to be based on (fc). In the latter the common factor is 
supposed to be a common object of perception inhabiting 
the "third" realm. In the former the common factor is not 
identified with an object of perception. (Physical objects 
are not reached until a much later stage of inference.) We 
use instead the group-structure of the resemblances (re- 
currencies) which by the theory of Chapter xn can be 
conceived to have an existence independent of that in which 
it is a structure. By this step we transfer the structure of 
individual experience into an external domain where it can 
be interwoven with the structures of other individual 
experiences similarly transferred. 

If the resemblances (b) were immediate data of experience 
something we were directly aware of there would be a 
justification for saying that the objects of perception, or 
sensory qualities, are less remote inferences than are physical 
objects. But it is obvious that the resemblances (a) are the 
only ones of which we are directly aware. Resemblances (b) 
are not only remote but very uncertain inferences from our 
experience, and they presuppose a detailed knowledge of 
physics and physiology. I cannot help thinking that your 
sensation when you are eating an apple resembles my sensa- 
tion in eating an apple more closely than it does my sensation 
in hearing a bell; but I do not know how to give any logical 
defence for this opinion because indeed resemblances of 
type (b) are so completely outside experience that we can 
form no idea of what such a resemblance would mean. 

When the philosopher proceeds further to associate 


the-taste-of-an-apple with an apple, he is attacking the pro- 
blem of finding the association between certain perceptions 
of taste, sight and touch which has also been attacked by the 
physicist, and more especially by the physiologist. If he 
neglects the experimental method of finding out the nature 
of the association, and propounds a hypothesis of their 
combined existence in an object outside the mind but directly 
apprehended by a number of minds a hypothesis directly 
contradictory to the scientific theory of the way in which 
the associated recurrence of sensations is determined it is 
inevitable that in die eyes of a scientist he should be classed 
with the idle speculators. 

I hope that this reply will not be looked on as an attempt 
to make public exposure of the hollow pretensions of realist 
philosophers. I am well aware that Prof. Stace cannot use 
all his armoury in a non-technical article. If sometimes he 
has not said quite what he means, I may still be to blame for 
not grasping his thought, seeing that I have not sufficient 
technical knowledge to follow the language of an exact 
statement. My intention has been to make clearer the case 
which philosophers have to meet, and to show that the new 
scientific philosophy is not quite the defenceless victim that 
some of them are apt to assume. 


I think that the fullest criticism of my scientific philosophy 
is contained in Mr C. E. M. Joad's Philosophical Aspects of 
Modern Science. I must first recognise the great care with 
which he has presented my arguments and conclusions, dis- 
tinguishing them where necessary from those of Jeans and 
Russell with which he also deals. Mr Joad belongs to the 
realist school, and therefore our controversy is to some extent 
the same as that discussed in the last section. But he appears 
to take a less extreme view than Prof. Stace. According to 


Stace "Chairs and tables and stars do really exist. They are 
exactly what they appear to be, coloured, spatial, resounding 
objects. Moreover, this familiar world is the only real world, 
the only world which really exists". Joad says (p. 12) 
"I cannot claim nor, indeed, did I expect that so far as 
the vindication of the common-sense world is concerned the 
attempt has met with much success. Chairs and tables do 
not, I fear, exist in the way in which in everyday life we 
suppose them to exist; but it does not follow that the 
physicist's analysis of them into atoms and electrons invested 
with secondary qualities projected upon them by the mind 
is therefore correct". And again (p. n) "The neo-realists' 
world of correlated sense data has borne less and less re- 
semblance to the common-sense world of physical objects 
which realism began by trying to preserve". 

Naturally this more moderate attitude brings him very 
much closer to my own view : 

I should hold, then, that the researches of the scientist are, 
equally with the perceptions of the plain man, the moral con- 
sciousness of the good man, the sensitivity of the artist and 
the religious experience of the mystic, revelatory of reality. 
Epistemologically they stand on equal terms. Such arguments 
as there are for supposing that any of these forms of experience 
is merely subjective, apply also to the others; but equally if any 
of them gives us information about a world external to ourselves, 
so also do the others. 

I might easily have mistaken that for an extract from my 
own writings. 

But the closeness of the approach makes me inclined all 
the more to question the need for a neo-realist's world which 
is neither the familiar world nor the scientific world nor a 
wider reality containing the scientific world as part of itself. 
When two men's paths are at right angles we suppose that 
they have different but equally valid objectives; when they 
ENPS 19 


diverge at a narrow angle we suspect that one of them may 
have mistaken the way. 

Mr Joad's criticisms are mainly directed to discovering 
inconsistencies of expressions and ideas in my writings. It 
is tempting to enter on a detailed defence; but it is perhaps 
better to confine myself to a general observation. I do not 
think that such discrepancies will appear so heinous to a 
scientist as they do to a philosopher. In science we do not 
expect finality. The theories described in the scientific part 
of this book do not form a complete and flawless system; 
there are incoherencies which we cannot remedy until further 
research gives us new light. It may well be that the scientific 
theory will be substantially modified in its future progress 
towards completion; nevertheless we feel justified in claiming 
that our present imperfect results embody a large measure 
of truth. I naturally look on scientific philosophy as subject 
to the same progressive advance. Undoubtedly the recent 
developments of physics have philosophical implications of 
the highest importance, and I have endeavoured to explain 
and elaborate diem. As with the scientific advances, so the 
philosophical advances can be consolidated into something 
like a system; but it does not disturb me unduly if there are 
loose ends that do not quite fit into the system glimpses of 
a deeper truth which we are not yet able to formulate. 

The advance of scientific philosophy has come from two 
main sources, the relativity theory and the quantum theory. 
When The Nature of the Physical World was written the 
scientific conceptions of the two theories were conflicting; 
and although there is now no longer a definite conflict the 
unification is still incomplete. It is not surprising that the 
philosophical outlook should display traces of the same dis- 
crepancy; but if the respective philosophies of relativity 
theory and quantum theory are not entirely harmonious they 
have at all events a large common denominator. 

I do not know whether it would be fair to say that the 


philosopher lays so much stress on formal consistency because 
he has little else by which to test the validity of his philo- 
sophy. But at any rate this does not apply to a philosophy 
developed on a scientific foundation. In science the unity 
and consistency of the system is an ideal to be reached by 
convergence. We are accustomed to finding different aspects 
of truth according to the way we approach it; we rejoice in 
its many-sidedness. I have been at no pains to suppress the 
many-sidedness of the truth which I believe is contained in 
the modern advance of physical science; and I therefore fall 
an easy victim to anyone who cares to collate passages in 
which I approach it from different angles. 

The shallower critics have also made capital by mixing 
passages in which the outlook is purely scientific, or passages 
in which I was leading the reader on, with those giving 
explicit statements of my philosophical ideas. In this respect 
Mr Joad has been entirely fair. It is necessary, however, to 
call attention to one surprising lapse. According to him, 
I affirm that atoms and electrons are objectively real and in 
fact that they (with other physical entities) constitute the 
sole objective reality. He asserts this twice (loc. cit. pp. 113, 
127) and on each occasion he quotes in support a sentence 
almost at the beginning of my book, "modern physics has 
assured me that my second, scientific table is the only one 
that is really there wherever 'there' may be". Surely the 
effect of the last four words is to suggest that there is a loop- 
hole, and that the assurance may appear in a different light 
when we discover what meaning (if any) is to be attached 
to "there". I certainly do not regard the entities of the 
physical world as the sole objective reality. As to whether 
atoms and electrons are objectively real, I divide my answer 
into two parts. Firstly, I do not think it is very important 
whether or not we use a particular phrase "objectively real", 
which nobody seems able to define. I have tried to explain 
the relation of atoms and electrons to the data of human 



experience. I think that the reader will be inclined to call 
whatever has this relation to experience "real"; but if he 
considers that it is an insufficient qualification for reality 
I shall not demur. It is purely a question of definition. 
Secondly, since atoms and electrons are the subject of 
quantum theory which is still in course of development, 
their scientific status is subject to some uncertainty, and this 
naturally affects their philosophical status. The effect of wave 
mechanics (especially as developed by Dirac) is to make the 
separation of the subjective and objective elements in human 
experience more indefinite. Relativity theory revealed an 
unsuspected subjective element in classical physics and cleared 
it away; wave mechanics has revealed a further subjective 
element; but its procedure is to let it stay and adopt methods 
suitable for treating a partially subjective world. So far as 
I can see, we find ourselves unable to reach by physical 
methods a purely objective world, and it would seem to 
follow that all the entities of physics have the partial sub- 
jectivity of the world to which they belong though, of 
course, they are not purely subjective. 

Turning to some of Joad's special criticisms, he objects 
(p. 31) that I represent the physical world as (a) abstracted 
by the mind of the scientist from a more comprehensive 
reality, (b) as constructed by the mind from relations and 
relata, and (c) given as embedded in a background of reality ; 
according to him it cannot be all three, i.e. abstracted, con- 
structed and given. But I see no reason why all three 
conceptions should not be applicable. The constellation Draco 
is a two-dimensional appearance of stars abstracted from their 
complete spatial distribution; it is constructed by a mind which 
is seeking resemblances to mythological characters and 
creatures; it is given as embedded in a galaxy of stars. If any 
one of these three aspects were missing the constellation 
Draco would not exist. Obviously it would not exist if 
there were no stars; there is no association between the stars 


composing it other than a fanciful resemblance to a serpent; 
and this resemblance only exists because it is contemplated 
as projected on the sky instead of in three dimensions. 
Similarly the world described by the equations of physics 
is, I believe, embedded in a background external to the 
individual mind, and constructed by a putting together 
of associations to which the mind is sensitive; its abstract 
character is obvious. 

There follow a number of criticisms (pp. 34-41) which 
suggest that Mr Joad has not grasped what is implied by the 
symbolic character of physical entities. It is as though, having 
said "Let x be the mass", I was supposed to be guilty of 
confusion in treating x both as an algebraic symbol and as a 
physical magnitude. Joad asks " What then is it that impinges 
on the sense organs to start the messages?" He is perplexed 
because the answer is atoms or things like atoms, which, 
1 have assured him, must not (in exact science) be thought 
of as possessing any other nature than that of a bundle of 
pointer-readings. How can a bundle of pointer-readings 
start a mental process? He might equally ask how can an 
algebraic symbol x make it difficult to shift an object? The 
answer is that the inertia or mass which makes the object 
difficult to move is symbolised by x. And similarly the 
bundles of pointer-readings symbolise the processes which 
start the messages. In particular the recurrencies of the 
pointer-readings stand for recurrencies of the processes. 

I pass on to the last of his formal criticisms. "The world 
of common experience is the datum from which the physicist 
starts and the criterion by which he judges the validity of 
the structure he raises. It is therefore presupposed as real 
and objective throughout."* 

* Since the context of this passage refers especially to my discussion 
of world building in which I stress the effect of mental selection on the 
characteristics of the physical world, it would be better to substitute 
"relevance" for "validity", i.e. relevance to the problem of experience. 


The argument appears to be that unless a datum is pre- 
supposed to be objective no inference can be based on it. 
This is so astonishing a suggestion that I wonder whether it 
can possibly be Mr Joad's real opinion. The data furnished 
by individual experience are clearly subjective, and it is 
ultimately from these data that the scientific conception of 
the universe is derived for what we term "collective ex- 
perience" is a synthesis of individual experiences. It would 
seem almost axiomatic that an ultimate datum is necessarily 
subjective. Joad then goes on to propound as a dilemma 
(p. 47) 

Thus atoms and quanta are the result of a process of inference 
based upon observation of the everyday world, while at the same 
time they originate a process which ends in the construction of 
the everyday world. Thus the everyday world must be pre- 
supposed before the process which results in its construction can 
take place. 

It seems a strange objection to scientific theory that it provides 
a universe capable of accounting for our everyday experience. 
Surely the whole intention of inference is that the result of 
the inference shall be that which is the origin of the datum 
from which the inference is made. When from an obser- 
vation of pink rats we infer the presence of alcohol, the 
validity of the inference lies in the fact that what we infer 
originates a process which ends in the mental construction of 
pink rats. Joad's dilemma seems to arise because he gratui- 
tously assumes the presupposition to be "presupposed as 
objectively rpal". But it is not presupposed that the pink 
rats are objectively real. 

His difficulty rather suggests that a cyclic scheme of know- 
ledge with which science has familiarised us is not yet 
appreciated in philosophy. I have formerly* illustrated the 
nature of a cyclic scheme by a revised version of "The House 

* The Nature of the Physical World, p. 262. 


that Jack Built" which instead of coining to an end repeats 
itself indefinitely " . . .that worried the cat, that killed the 
rat, that ate the malt, that lay in the house, that was built 
by the priest all shaven and shorn, that married . . .". Wherever 
we start in the cycle we presuppose something that we reach 
again by following round the cycle. The scheme of physics 
constitutes such a cycle; and equally we may contemplate 
a wider cycle embracing that which is beyond physics. 
Starting at the point of the cycle which corresponds to our 
individual perceptions, we reach other entities which are 
constructs from our perceptions. From these we reach other 
entities, and so on for a number of steps. When we seem to 
have travelled a long way from our starting point, we find 
that our perceptions (or more strictly the recurrencies in our 
perceptions) reappear as constructs from the last-reached 
entities. The fact that we return by a circuit and not by 
retracing our steps secures that our adventure is an extension 
of knowledge and not an excursion in tautology. By the 
method of Chapter xii we can extract the group structure 
from the cycle and so express the same truth symbolically 
without a formal presupposition if we prefer. 


The burning question of Determinism is a source of much 
criticism and controversy. Although the controversial side 
of the subject is not neglected in Chapter iv, it may be of 
interest to defend my position with more explicit reference 
to the views and statements of leading determinists, especially 
Planck and Einstein. The case against me, based mainly on 
these authorities, has been ably stated by Sir Herbert Samuel ;* 
the following is taken from an article which I wrote in reply, f 

* "Cause, Effect, and Professor Eddington", The Nineteenth Century 
and After, April 1933. 
f Ibid. June 1933. Reproduced by kind permission of the Editor. 


Sir Herbert Samuel has arrested me for trying to rob the 
public of their most valuable beliefs, and he has placed in the 
witness-box three of the most eminent physicists now living 
to give evidence for the prosecution. I suspect that he counts 
more on the impression that will be produced by this array 
of authority than on the actual content of their evidence; for 
there is more protestation than argument in what they have 
to say. So far as authority is concerned, it would scarcely 
be possible to name a more formidable trio than Planck, 
Einstein, Rutherford; nevertheless, I trust that the jury before 
reaching their verdict will hear patiently what I shall say in 
my defence. 

The occasion of the trial is that I (in common with many 
modern physicists) have disseminated unbelief in the "Prin- 
ciple of Causality'*, better known to the public as the 
doctrine of Determinism. The first designation is generally 
used by Sir Herbert and his witnesses, but I am not sure that 
it will be understood by the general reader. I hope the 
language of the indictment will not lead anyone to suppose 
that I deny that effects may proceed from causes. The 
assembly of spectators at an international football match is 
undoubtedly a cause of the congestion in the streets of 
Twickenham an hour or so later. But what the principle of 
causality asserts is that observed causation of this kind is 
analysable indefinitely, so that each minute movement in the 
crowd was likewise determined in advance by causes existing 
hours (or centuries) before. It is this exact and universal 
causality or predetermination that I challenge. 

Of the three witnesses Prof. Max Planck is the one on 
whom my accuser chiefly relies, and he is the only one whose 
evidence is in a form which admits of detailed examination. 
Rutherford, indeed, is too wary to enter into a discussion 
which might savour of philosophy and takes refuge in a 
platitude which, though presumably meant for a condemna- 
tion of indeterminists, can equallv be read as a condemnation 


of determinists. Planck's views are of special interest because 
he is the founder of the physical theory which has led to the 
present crisis; and his arguments are contained in a carefully 
written book. 

In the controversy Determinism v ersus Indeterminism it is 
essential to have a clear understanding on which side the onus 
of proof lies which side is putting forward a positive doctrine 
which it wishes the other side to embrace. Sir Herbert quotes 
a letter from Lord Rutherford which says : 

It seems to me unscientific and also dangerous to draw far-flung 
deductions from a theoretical conception which is incapable of 
experimental verification, either directly or indirectly. 

To which side does this apply? My case against Sir Herbert 
Samuel and his fellow-determinists has been that they de- 
velop a far-reaching philosophical outlook based on the 
principle of causality a principle which has not been ex- 
perimentally verified. Here is Einstein's testimony:* 

Hitherto people have looked upon the Principle of Causality 
as a proposition which would in the course of years admit of 
experimental proof with an ever-increasing exactitude. Positively 
defined as a limiting proposition, the principle runs as follows 

Now Heisenberg has discovered a flaw in the proposition 

The principle of causality loses its significance as an empirical 

Causality is thus only conceivable as a Form of the theoretical 
system. Now modern physicists are mainly of the opinion that 
it is inadmissible to build up any sort of theory on what cannot, 
in principle, be tested. 

Einstein, it will be seen, admits that the principle of 
causality is a positive proposition. He makes no pretence 
that it has been experimentally verified. Having lost its 
empirical significance, it is out of range of experimental test 

* From a letter to Sir Herbert Samuel, published in Sir Herbert' 
Presidential Address: Philosophy and the Ordinary Man, p. 15. 


and is indeed only conceivable as a form of theoretical system. 
The words of Lord Rutherford recoil on the prosecution like 
a boomerang. Out of the mouth of their own witness the 
principle of causality the valuable belief of which I am 
accused of robbing the public is shown to be "a theoretical 
conception which is incapable of experimental verification". 

Further, compare Planck's testimony:* 

Is it perfectly certain and necessary for human thought that for 
every event in every instance there must be a corresponding 
cause?. . . Of course the answer is in the negative 

Thus from the outset we can be quite clear about one very 
important fact, namely, that the validity of the law of causation 
for the world of reality is a question that cannot be decided on 
grounds of abstract reasoning. 

with Einstein's testimony :f 

Look here. Indeterminism is quite an illogical concept If 
I say that the average life-span of such an atom is indetermined 
in the sense of not being caused, then I am talking nonsense. 

Gentlemen of the jury, you have been assured by Planck 
that it is not a logical necessity of human thought that every 
event should have a corresponding cause, but nevertheless 
"physical science, together with astronomy and chemistry 
and mineralogy, are all based on the strict and universal 
validity of the principle of causality "4 Einstein tells you 
that denial of the principle is illogical, and that it is nonsense 
to speak of an event as not having a cause; but the principle 
of causality is a theoretical proposition which, by its very 
nature, is incapable of experimental test. Rutherford warns 
you that it is unscientific to base your conclusions on a 
theoretical conception which is incapable of experimental 
verification. So now you know just what you are to think 

* Where is Science Going? pp. 112, 113. 

I Ibid. p. 202. 
Ibid. p. 147- 


of the principle of causality, according to the voice of 

Since Planck's discussion is the most extensive, I will treat 
him as the main witness. I think it is significant of his 
attitude that he devotes a whole chapter of his book to a 
survey of the views of different schools of philosophers, 
whereas the results of physics are accorded less than five 
pages.* These claim to give the " answer of physics". The 
crucial paragraph is one already quoted by Sir Herbert 

In point of fact, statistical laws are dependent on the assumption 
of the strict law of causality functioning in each particular case. 
And the non-fulfilment of the statistical rule in particular cases 
is not therefore due to the fact that the law of causality is not 
fulfilled, but rather to the fact that our observations are not 
sufficiendy delicate and accurate to put the law of causality to a 
direct test in each case. If it were possible to follow the move- 
ment of each individual molecule in this very intricate labyrinth 
of processes, then we should find in each case an exact fulfilment 
of the dynamical laws. 

How does Prof. Planck know this? He speaks as though 
the whole course of Nature lay revealed to him. Although 
we cannot apply the test, he knows that the test would be 
exactly fulfilled if we could apply it. He omits to tell the 
reader that there is no mention in any modern treatise on 
quantum theory of die dynamical laws (i.e. causal, as distinct 
from statistical, laws) to which he here alludes, for the reason 
that they have not been discovered or even guessed at. Prof. 
Planck is at liberty to bring this view forward as a hypothesis 
(if he is prepared to risk Lord Rutherford's displeasure) ; it is, 
in fact, the hypothesis usually made by determinists in order 
to render their doctrine tenable. But to announce it as the 
answer of science is surely a grave misstatement. Actually 
the present trend of physical science is against it. I do not 

* Ibid. pp. 143-147- 


mean that it lias been disproved; but phenomena which were 
formerly thought to be a direct consequence of particular 
causal laws are now acknowledged to be the result of 
statistical laws, so that they no longer constitute support for 
Planck's contention. Evidence formerly trumpeted as favour- 
able is now found to be indifferent. 

Possibly Prof. Planck intended to stress the first sentence 
in the above quotation, meaning thereby that it has been 
proved (mathematically or logically) that unless each in- 
dividual is governed by strict causal law statistical laws for 
the assembly are impossible. But does he seriously expect 
us to believe that die regular experience of life assurance 
companies would be impossible if the individuals insured had 
any genuine free will? I think not. I think he is merely 
stating a practice which used to be followed of formulating 
a system of causal law before deducing statistical laws 
forgetting that Heisenberg, Schrodinger, Dirac, and others 
have abandoned this procedure, and that it is their statistical 
laws which are the basis of existing quantum theory. 

Sir Herbert asks whether I am "justified in saying, not 
that certain scientists, but that science itself, has abandoned 
determinism". I am glad he stresses this distinction. It is 
illustrated by another of his quotations from Planck: 

Some essential modification seems to be inevitable; but I firmly 
believe, in company with most physicists, that the quantum 
hypothesis will eventually find its exact expression in certain 
equations which will be a more exact formulation of the law of 

Thus the causal law is to be found, not in the quantum theory 
as it is, but in what Planck believes it will eventually become. 
That is just what I maintain. The law of causality does not 
exist in science to-day in that body of systematic knowledge 
and hypothesis which has been experimentally confirmed, 
tt exists only in the anticipations of certain scientists anti- 


cipations which naturally are coloured by their philosophical 

The philosophical chapter in Planck's book contains one 
feature which very much concerns our discussion. The 
chapter begins:* 

This is one of man's oldest riddles. How can the independence 
of human volition be harmonized with the fact that we arc 
integral parts of a universe which is subject to the rigid order of 
Nature's laws? 

At first sight these two aspects of human existence seem to be 
logically irreconcilable. On the one hand we have the fact that 
natural phenomena invariably occtir according to the rigid 
sequence of cause and effect. This is the indispensable postulate 
of all scientific research.. . .But on the other hand we have our 
most direct and intimate source of knowledge, which is the human 
consciousness, telling us that in the last resort our thought and 
volition are not subject to this causal order. 

The whole chapter is occupied with the various attempts to 
solve this riddle. 

Obviously the riddle does not arise unless we accept the 
law of causality in Nature. There may be other aspects of the 
problem of free will leading to other riddles; but the main 
dilemma, which Planck places in the forefront of the problem, 
ceases to exist. Many writers have said that our researches 
into the laws of atomic physics have no bearing on the 
problem of free will and volition. Planck evidently is not of 
this way of thinking. For him, as for me, the main problem 
turns on whether physics does or does not assert the principle 
of causality. 

It is on this point that a number of popular scientific writers 
have taken up a position that seems to me preposterous. They 
hold that, since strict causality has not been disproved, and is 
not incompatible with the new theories, there has not been 
any modification of the problem. But the dilemma can only 

* Where is Science Going? p. 107. 


be created if physics gives positive support to the principle of 
causality. It takes two combatants to make a fight not one 
combatant and one neutral. 

In the present controversy there has been a great tendency 
to confuse two questions "Is the law of causality true of the 
physical universe?" and "Is it the present basis of physical 
science?" I have quoted Prof. Planck's picture of the 
niedianism of Nature, which obviously goes far beyond 
anything warranted by existing knowledge. If I declared 
this picture to be untrue, I should be open to the same charge 
of dogmatism as he is. But I can say most assuredly that this 
picture is not the basis of present-day physics. Present-day 
physics is simply indifferent to it. We might believe in it 
to-day and disbelieve in it to-morrow; not a symbol in the 
modern textbooks of physics would be altered. 

Einstein (unlike Planck) fully recognises this change. 
Whereas Planck holds that modern physics is still based on 
the law of causality, Einstein recognises that it is not, and 
he deplores the change. It may be added that under this con- 
viction Einstein has for several years been actively engaged 
in search for a new theory which shall restore the law of 
causality to its old supreme position; but hitherto he has not 
been successful. I need scarcely say that a writer who deals 
with the philosophical implications of physical science must 
base his assertions on the existing scheme of knowledge 
which has resulted from the exertions of Planck, Einstein, 
Rutherford and others, not on a theory which Einstein hopes 
some day to produce. 

I do not think that the social and political consequences of 
my teaching will be so terrible as Sir Herbert Samuel fears. 
He suggests that a student of mine, learning that if he sets 
light to a barrel of gunpowder an explosion, although highly 
probable, is not certain, may decide to put the matter to the 
test. The result will doubtless bear out my assertion that an 
explosion was exceedingly probable, so I do not see where 


the grievance of the relatives of the deceased comes in. And 
at least I concede to my student freedom to avoid the 
catastrophe by abstaining from acting in this strange manner; 
whereas, according to Sir Herbert Samuel and the deter- 
minists, the explosion of the barrel is the inevitable outcome 
of causes which have existed from the beginning of time. 

Nor do I think that the substitution of high probability 
for certainty in the political and economic sphere will be 
disastrous. It would seem that at the present moment my 
opportunity for destroying "certainty" in political and 
economic science is rather limited. Might it not then be 
better to stress the other side of my conclusions that, so 
far as is known, our future is not wholly prearranged by 
physical law? It is we who have to shape it for better or 
worse. I have on occasion supported Sir Herbert Samuel and 
voted for his political efforts for amelioration. My decision 
was on probability; I could not expect complete certainty 
that his policy would achieve its end. If any of our leaders 
can offer the world a solution of the present troubles, we shall 
not ask for certainty; let him but convince us that the prob- 
ability of success is shall we say? a million to one, and we 
will follow him to the last ditch. 


It would be of little advantage to discuss here the contro- 
versial aspects of the conclusions which I have reached, or 
have accepted from the work of others, in regard to purely 
scientific problems. So far as space permits, I have tried to 
meet in advance the objections most likely to occur to an 
attentive reader. But if an expert colleague is unconvinced, 
or claims to have discovered mistakes and fallacies, the right 
place to meet him is in a technical journal where matne- 
matical formulae can be countered by mathematical formulae 
and all our resources for the discovery of truth can be brought 


to bear. Those who are not prepared to study for themselves 
the technical arguments, must make what they can of rival 
assurances that "Codlin's the friend, not Short". I can do 
no more than pass on such glimpses of illumination as I have 
found in my own efforts to understand. 

There is a point connected with popular expositions of 
physical science which is perhaps not generally realised. As a 
rule the results which they translate into non-technical lan- 
guage are obtained partly by strict mathematical deduction, 
and partly by general arguments as to what hypotheses seem 
best to accord with physical observation and experiment. 
Now physics is, or should be, undogmatic; mathematics is, 
and must be, dogmatic. No mathematician is infallible; he 
may make mistakes; but he must not hedge. Even in this age 
which dislikes dogma, there is no demand for an undogmatic 
edition of Euclid; and the examinee who was unable to 
prove the binomial theorem but "thought he had made it 
rather plausible " is not held up as an example to be followed. 
In summarising conclusions for the general reader, mathe- 
matical and physical considerations become fused together, 
and it is impossible to show without elaboration of technical 
detail where the dogmatic mathematical deduction ends and 
the plausible physical inference begins. You may therefore 
find that a book which on the whole reflects the liberal un- 
dogmatic attitude of science is chequered with pronounce- 
ments which suggest omniscience and intolerance. The latter 
are a sign (or so it is charitable to assume) that the argument 
has shifted into the region of strict mathematical deduction, 
where hedging is not permitted and a definite lead must be 

Correspondingly there are two kinds of criticism. The one 
claims to have found a flaw in an author's mathematical 
deduction; the other dissents from his judgment of the 
evidence. As to the former, we can only say that one of the 
parties must be culpably wrong. Supposing, however, that 


there is agreement on the mathematical side of the problem, 
there is often room for interesting and valuable controversy 
on questions of judgment; and some divergence of view is 
beneficial. Where judgment is more than usually difficult 
I have tried to indicate the corresponding uncertainty; but 
there is scarcely any physical conclusion which we can hold 
as safe from all possibility of revision. Even such a funda- 
mental law as the conservation of energy is now being 
challenged on account of certain phenomena observed in the 
production of j3 rays; I do not myself believe that it is in 
serious danger, but perhaps I am wrong. On the subject of 
the constitution of the stars we can scarcely doubt that 
substantial knowledge has been gained, consideration having 
been given to all contingencies which we should deem 
reasonably likely; nevertheless few, if any, of the accepted 
conclusions, either for the deep interior or for the surface 
layers of a star, are so unconditional that a star might not 
evade them if it really wanted to be nasty. 

No doubt a detached critic would often recommend sus- 
pension of judgment on questions as to which I have ventured 
to adopt a definite opinion. But I think it would give a 
wrong picture of scientific activity to view it entirely through 
such a critic's eyes. The working scientist, like any other man 
who wishes to accomplish something, must steer a middle 
course between chronic indecision and precipitant judgment. 
It is not just a question whether he shall believe this or believe 
that; it is a choice which may determine whether or not 
several years of his life shall be spent in working along a 
blind alley. 


My last round will be with Bertrand Russell. I think that he 

more than any other writer has influenced the development 

of my philosophical views; and my debt to him is great 

ENPS 20 


indeed. But this is necessarily a quarrelsome chapter, and 
I must protest against the following accusation * 

Sir Arthur Eddington deduces religion from die fact that atoms 
do not obey the laws of mathematics. Sir James Jeans deduces 
it from the fact that they do. 

Russell here attributes to me a view of the basis of religion 
which I have strongly opposed whenever I have touched on 
the subject. I gather from what precedes this passage that 
Russell is really referring to my views on free will, which he 
appears to regard as equivalent to religion; but even so the 
statement is far from true. I have not suggested that either 
religion or free will can be deduced from modern physics; 
I have limited myself to showing that certain difficulties in 
reconciling them with physics have been removed. If I found 
a prevailing opinion that Russell could not be a competent 
mathematician because he had claimed to square the circle, 
I might, in defending him, point out that the report that he 
had made such a claim was without foundation. Would it 
be fair to say that I deduce that Russell is a competent 
mathematician from the fact that he has not claimed to 
square the circle? 

One might have regarded the foregoing as a casual sacrifice 
of accuracy to epigram, but other passages make the same 
kind of accusationrf 

It will be seen that Eddington, in this passage, $ does not infer 
a definite act of creation by a Creator. His only reason for not 
doing so is that he does not like the idea. The scientific argument 
leading to the conclusion which he rejects is much stronger than 
the argument in favour of free will, since that is based on 
ignorance, whereas the one we are now considering is based upon 
knowledge. This illustrates the fact that the theological conclusions 
drawn by scientists from their science are only such as please 

* The Scientific Outlook, p. 112. 
Ibid. p. 121. 
The Nature of the Physical World, p. 83. 


them, and not such as their appetite for orthodoxy is insufficient 
to swallow, although the argument would warrant them. 

And again (p. 96) : 

[Eddington's] optimism is based upon the time-honoured 
principle that anything which cannot be proved untrue may be 
assumed to be true, a principle whose falsehood is proved by the 
fortunes of bookmakers. 

Neither my optimism, nor my belief in free will and in 
religion, nor my belief in Russell's competence as a mathe- 
matician is based on this time-honoured principle. But 
however strong may be the positive grounds for one's 
opinions, it is not irrelevant to examine the negative grounds 
and satisfy oneself and others that the evidence which seemed 
hostile to these beliefs has collapsed. 

Memories arc short, and one man is sometimes saddled 
with another man's opinions. It seems worth while therefore 
to give quotations showing how completely Russell has mis- 
stated my view of the relation of science and religion. I think 
that every book or article in which I have touched on religion 
is represented in these extracts, except an early essay (i9 2 5) 
which does not provide a passage compact enough to quote. 

The starting-point of belief in mystical religion is a conviction 
of significance or, as I have called it earlier, the sanction of a 
striving in the consciousness. This must be emphasised because 
appeal to intuitive conviction of this kind has been the foundation 
of religion through all ages and I do not wish to give the im- 
pression that we have now found something new and more 
scientific to substitute. I repudiate the idea of proving the 
distinctive beliefs of religion either from the data of physical 
science or by the methods of physical science. (The Nature of the 
Physical World, p. 333.) 

The lack of finality of scientific theories would be a very 
serious limitation of our argument, if we had staked much on 
their permanence. The religious reader may well be content that 
I have not offered him a God revealed by the quantum theory, and 



therefore liable to be swept away in the next scientific revolution. 
(Ibid. p. 353-) 

It is probably true that the recent changes of scientific thought 
remove some of the obstacles to a reconciliation of religion with 
science; but this must be carefully distinguished from any pro- 
posal to base religion on scientific discovery. For my own part 
I am wholly opposed to any such attempt. (Science and the 
Unseen World, p. 45.) 

The passages quoted by Mr Cohen make it clear that I do not 
suggest that the new physics "proves religion" or indeed gives 
any positive grounds for religious faith. But it gives strong 
grounds for an idealistic philosophy which, I suggest, is hospitable 
towards a spiritual religion, it being understood that the guest 
must provide his own credentials. In short the new conception 
of the physical universe puts me in a position to defend religion 
against a particular charge, viz. the charge of being incompatible 
with physical science. It is not a general panacea against atheism. 
If this is understood,. . .it explains my "great readiness to take 
the present standing of certain theories of physics as being final"; 
anybody can defend religion against science by speculating on 
the possibility that science may be mistaken. It explains why I 
sometimes take the essential truth of religion for granted; die 
soldier whose task is to defend one side of a fort must assume 
that the defenders of the other side have not been overwhelmed. 
(Article in The Freethinker.) 

I now turn to the question, what must be put into the skeleton 
scheme of symbols. I have said that physical science stands aloof 
from this transmutation, and if I say anything positive on this 
side of the question it is not as a scientist that I claim to speak. 
(Broadcast Symposium, Science and Religion.) 

The bearing of physical science on religion is that the 
scientist has from time to rime assumed the duty of signalman 
and set up warnings of danger not always unwisely. If 
I interpret the present situation rightly, a main-line signal 
which had been standing at danger has now been lowered. 
But nothing much is going to happen unless there is an 


Modern science, in so far as I am familiar with it through my own scientific 
work, mathematics and physics make the world appear more and more 
as an open one, as a world not closed but pointing beyond itself. . . . 
Science finds itself compelled, at once by the epistemological, the physical 
and the constructive-mathematical aspect of its own methods and results, 
to recognise this situation. It remains to be added that science can do no 
more than show us this open horizon; we must not by including the 
transcendental sphere attempt to establish anew a closed (though more 
comprehensive) world. HERMANN WEYL, The Open World. 


OUR home, the Earth, is the fifth or sixth largest planet 
belonging to a middle grade star in the Milky Way. Within 
our galaxy alone there are perhaps a thousand million stars 
as large and as luminous as the sun; and this galaxy is one 
of many millions which formed part of the same creation 
but are now scattering apart. Amid this profusion of worlds 
there are perhaps other globes that are or have been inhabited 
by beings as highly developed as Man; but we do not think 
they are at all common. The present indications seem to be 
that it is very long odds against a particular star undergoing 
the kind of accident which gave birth to the solar system. 
It seems that normally matter collects in big masses with 
excessively high temperature, and the formation of small 
cool globes fit for habitation is a rare occurrence. Nature 
seems to have been intent on a vast evolution of fiery worlds, 
an epic of milliards of years. As for Man it seems unfair 
to be always raking up against Nature her one little in- 
advertence. By a trifling hitch of machinery not of any 
serious consequence in the development of the universe 


some lumps of matter of the wrong size have occasionally 
been formed. These lack the purifying protection of intense 
heat or the equally efficacious absolute cold of space. Man 
is one of the gruesome results of this occasional failure of 
antiseptic precautions. 

To realise the insignificance of our race before the majesty 
of the universe may be healthful; but it brings to us an 
alarming thought. For Man is the typical custodian of certain 
qualities or illusions, which make a vital difference to the 
significance of things. He displays purpose in an inorganic 
world of chance. He can represent truth, righteousness, 
sacrifice. In him there flickers for a few brief years a spark 
from the divine spirit. Are these of as little account in the 
universe as he is ? 

It may be going too far to say that our bodies are pieces 
of stellar matter which, by a contingency not sufficiently 
guarded against in Nature, have evaded the normal destiny, 
and have taken advantage of low temperature conditions to 
assume unusual complication and perform the series of antics 
we call "life". I neither assert nor deny this view; but I 
regard it as so much of an open question that I am unwilling 
to base my philosophy or my religion on the assumption 
that it must necessarily break down. But there is another 
approach to the problem. Science is an attempt to read the 
cryptogram of experience; it sets in order the facts of sensory 
experience of human beings. Everyone will agree that this 
attempt has met with considerable success; but it does not 
start quite at the beginning of the Problem of Experience, 
The first question asked about scientific facts and theories, 
such as we have been discussing in this book, is "Are they 
true?" I would emphasise that even more significant than 
the scientific conclusions themselves is the fact that this 
question so urgently arises about them. The question "Is it 
true?" changes the complexion of the world of experience 
not because it is asked about the world, but because it is 


asked in the world. When we go right back to the beginning, 
the first thing we must recognise in the world of experience 
is something intent on truth something to which it matters 
intensely that beliefs should be true. This is no elusive 
cryptogram; it is not written in the symbolic language in 
which we describe the unknowable activities of unknown 
agents in the physical universe. Before we invite science to 
take the problem in hand and put in order the facts of sensory 
experience, we have settled the first ingredient of the world 
of experience. If science in its survey rediscovers that in- 
gredient, well and good. If not, then science may claim to 
account for the universe, but what is there to account for 

What is the ultimate truth about ourselves? Various 
answers suggest themselves. We are a bit of stellar matter 
gone wrong. We are physical machinery puppets that strut 
and talk and laugh and die as the hand of time pulls the 
strings beneath. But there is one elementary inescapable 
answer. We are that which asks the question. Whatever else 
there may be in our nature, responsibility towards truth is 
one of its attributes. This side of our nature is aloof from the 
scrutiny of the physicist. I do not think it is sufficiently 
covered by admitting a mental aspect of our being. It has to 
do with conscience rather than with consciousness. Concern 
with truth is one of those things which make up the spiritual 
nature of Man. There are other constituents of our spiritual 
nature which are perhaps as self-evident; but it is not so easy 
to force an admission of their existence. We cannot recognise 
a problem of experience without at the same time recognising 
ourselves as truth-seekers involved in the problem. The 
strange association of soul and body of responsibility 
towards truth with a particular group of carbon compounds 
is a problem in which we naturally feel intense interest; 
but it is not an anxious interest, as though the existence of 
a spiritual significance of experience were hanging in the 


Balance, That significance is to be regarded rather as a datum 
of the problem; and the solution must fit the data; we must 
not alter the data to fit an alleged solution. 

I do not regard the phenomenon of life (in so far as it can 
be separated from the phenomenon of consciousness) as 
necessarily outside the scope of physics and chemistry. 
Arguments that because a living creature is an organism it 
ipso facto possesses something which can never be understood 
in terms of physical science do not impress me. I think it is 
insufficiently recognised that modern theoretical physics is 
very much concerned with the study of organisation; and 
from organisation to organism does not seem an impossible 
stride. But equally it would be foolish to deny the magnitude 
of the gulf between our understanding of the most complex 
form of inorganic matter and the simplest form of life. Let 
us suppose, however, that some day this gulf is bridged, and 
science is able to show how from the entities of physics 
creatures might be formed which are counterparts of our- 
selves even to the point of being endowed with life. The 
scientist will perhaps point out the nervous mechanism of the 
creature, its powers of motion, of growth, of reproduction, 
and end by saying "That's you". But it has yet to satisfy 
the inescapable test. Is it concerned with truth as I am ? Then 
I will acknowledge that it is indeed myself. The scientist 
might point to motions in the brain and say that these really 
mean sensations, emotions, thoughts; and perhaps supply a 
code to translate the motions into the corresponding thoughts. 
Even if we could accept this inadequate substitute for con- 
sciousness as we intimately know it, we must still protest: 
"You have shown us a creature which thinks and believes; 
you have not shown us a creature to whom it matters that 
what it thinks and believes should be true". The inmost ego, 
possessing what I have called the inescapable attribute, can 
never be part of the physical world unless we alter the 
meaning of the word 'physical" o as to be synonymous 


with " spiritual" a change scarcely to the advantage of 
clear thinking. But having disowned our supposed double, 
we can say to the scientist: "If you will hand over this Robot 
who pretends to be me, and let it be filled with the attribute 
at present lacking and perhaps other spiritual attributes which 
I claim as equally self-evident, we may arrive at something 
that is indeed myself". 

A few years ago the suggestion of taking the physically 
constructed man and adapting him to a spiritual nature by 
casually adding something, would have been a mere figure 
of speech a verbal gliding over of insuperable difficulties. 
In much the same way we talk loosely of constructing a 
Robot and then breathing life into it. A Robot is pre- 
sumably not constructed to bear such last-minute changes 
of design; it is a delicate piece of mechanism made to work 
mechanically, and to adapt it to anything else would involve 
entire reconstruction. To put it crudely, if you want to fill 
a vessel with anything you must make it hollow, and the 
old-fashioned material body was not hollow enough to be 
a receptacle of mental or of spiritual attributes. The result 
was to place consciousness in the position of an intruder in 
the physical world. We had to choose between explaining 
it away as an illusion or perverse misrepresentation of what 
was really going on in the brain, and admitting an extraneous 
agent which had power to suspend the regular laws of Nature 
and asserted itself by brute interference with the atoms and 
molecules in contact with it. 

Our present conception of the physical world is hollow 
enough to hold almost anything. I think the reader will 
agree. There may indeed be a hint of ribaldry in his hearty 
assent. What we are dragging to light as the basis of all 
phenomena is a scheme of symbols connected by mathe- 
matical equations. That is what physical reality boils down 
to when probed by the methods which a physicist can apply. 
A skeleton scheme of symbols proclaims its own hollowness. 


It can be nay it cries out to be filled with something that 
shall transform it from skeleton into substance, from plan 
into execution, from symbols into an interpretation of the 
symbols. And if ever the physicist solves the problem of the 
living body, he should no longer be tempted to point to his 
result and say "That's you". He should say rather "That is 
the aggregation of symbols which stands for you in my 
description and explanation of those of your properties 
which I can observe and measure. If you claim a deeper 
insight into your own nature by which you can interpret 
these symbols a more intimate knowledge of the reality 
which I can only deal with by symbolism you can rest 
assured that I have no rival interpretation to propose". The 
skeleton is the contribution of physics to the solution of the 
Problem of Experience; from the clothing of the skeleton 
it stands aloof. 


The scientific conception of the world has come to differ 
more and more from the commonplace conception until we 
have been forced to ask ourselves what really is the aim of 
this scientific transmutation. The doctrine that things are not 
what they seem is all very well in moderation; but it has 
proceeded so far that we have to remind ourselves that the 
world of appearances is the one to which we have actually 
to adjust our outward lives. That was not always so. At first 
the progress of scientific thought consisted in correcting gross 
errors in the familiar conception of things. We learned that 
the earth was spherical, not flat. That does not refer to some 
abstract scientific earth, but to the homely earth that we 
know so well. I do not think any of us have any difficulty 
in picturing the earth as spherical. I confess that the idea is 
so familiar to me that it obtrudes itself irrelevantly, and I am 
liable to visualise a Test Match in Australia as being played 
Upside down. We learned that the earth was rotating. For 


the most part we give an intellectual assent to this conclusion 
without attempting to weave it into our familiar conception; 
but we can picture it if we try. In Rossetti's poem the Blessed 
Damosel looked down from the golden balcony of Heaven 

The void, as low as where this earth 
Spins like a fretful midge. 

Looking from the abode of truth, perfect truth alone can 
enter her mind. She must see the earth as it really is like 
a whirling insect. But now let us try her with something 
fairly modern. In Einstein's theory the earth, like other 
matter, is a curvature of space-time, and what we commonly 
call the spin of the earth is the ratio of two of the components 
of curvature. What is the Blessed Damosel to make of that? 
I am afraid she will have to be a bit of a blue-stocking. 
Perhaps there is no great harm in that. I am not sure that 
I would think it derogatory to an angel to accuse him of 
understanding Einstein's theory. My objection is more 
serious. If the Blessed Damosel sees the earth in the Ein- 
steinian way she will be seeing truly I can feel little doubt 
as to that but she will be missing the point. It is as though 
we took her to an art gallery, and she (with that painful 
truthfulness which cannot recognise anything that is not 
really there) saw ten square yards of yellow paint, five of 
crimson, and so on. 

So long as physics in tinkering with the familiar world 
was able to retain those aspects which appeal to the aesthetic 
side of our nature, it might with some show of reason make 
claim to cover the whole of experience; and those who 
claimed that there was another, religious aspect of our 
existence had to fight for their claim. But now that its 
picture omits so much that is obviously significant, there is 
no suggestion that it is the whole truth about experience, 
To make such a claim would bring protest not only from 


the religiously minded but from all who recognise that Man 
is not merely a scientific measuring machine. 

Physics provides a highly perfected answer to one specialised 
problem which confronts us in experience. I do not wish to 
minimise the importance of the problem and the value of 
the solution. We have seen (p. n) how in order to focus 
the problem the various faculties of the observer have been 
discarded, and even his sensory equipment simplified, until 
the problem becomes such as our methods are adequate to 
solve. For the physicist the observer has become a symbol 
dwelling in a world of symbols. But before ever we handed 
over the problem to the physicist we had a glimpse of Man 
as a spirit in an environment akin to his own spirit. 

In so far as I refer in these lectures to an experience reaching 
beyond the symbolic equations of physics I am not drawing 
on any specialised scientific knowledge; I depend, as anyone 
might do, on that which is the common inheritance of human 

We recognise that the type of knowledge after which 
physics is striving is much too narrow and specialised to con- 
stitute a complete understanding of the environment of the 
human spirit. A great many aspects of our ordinary life and 
activity take us outside the outlook of physics. For the most 
part no controversy arises as to the admissibility and im- 
portance of these aspects; we take their validity for granted 
and adapt our life to them without any deep self-questioning. 
Any discussion as to whether they are compatible with the 
truth revealed by physics is purely academic; for whatever 
the outcome of the discussion, we are not likely to sacrifice 
them, knowing as we do at the outset that the nature of Man 
would be incomplete without such outlets. It is therefore 
somewhat of an anomaly that among the many extra- 
physical aspects of experience religion alone should be singled 
out as specially in need of reconciliation with the knowledge 
contained in science. Why should anyone suppose that all 


that matters to human nature can be assessed with a measuring 
rod or expressed in terms of the intersections of world-lines? 
If defence is needed, the defence of a religious outlook must, 
I think, take the same form as the defence of an aesthetic 
outlook. The sanction seems to He in an inner feeling of 
growth or achievement found in the exercise of the aesthetic 
faculty and equally in the exercise of the religious faculty. 
It is akin to the inner feeling of the scientist which persuades 
him that through the exercise of another faculty of the mind, 
namely its reasoning power, we reach something after which 
the human spirit is bound to strive. 

It is by looking into our own nature that we first discover 
the failure of the physical universe to be co-extensive with 
our experience of reality. The "something to which truth 
matters" must surely have a place in reality whatever 
definition of reality we may adopt. In our own nature, or 
through the contact of our consciousness with a nature 
transcending ours, there are other things that claim the same 
kind of recognition a sense of beauty, of morality, and 
finally at the root of all spiritual religion an experience which 
we describe as the presence of God. In suggesting that these 
things constitute a spiritual world I am not trying to sub- 
stantialise them or objectivise them to make them out other 
than we find them to be in our experience of them. But 
I would say that when from the human heart, perplexed with 
the mystery of existence, the cry goes up, " What is it all 
about?" it is no true answer to look only at that part of 
experience which comes to us through certain sensory organs 
and reply: "It is about atoms and chaos; it is about a universe 
of fiery globes rolling on to impending doom; it is about 
tensors and non-commutative algebra". Rather it is about 
a spirit in which truth has its shrine, with potentialities of 
self-fulfilment in its response to beauty and right. Shall I not 
also add that even as light and colour and sound come into 
our minds at the prompting of a world beyond, so these 


other stirrings of consciousness come from something which, 
whether we describe it as beyond or deep within ourselves, 
is greater than our own personality? 

It is the essence of religion that it presents this side of 
experience as a matter of everyday life. To live in it, we have 
to grasp it in the form of familiar recognition and not as a 
series of abstract scientific statements. The man who com- 
monly spoke of his ordinary surroundings in scientific 
language would be insufferable. If God means anything in 
our daily lives, I do not think we should feel any disloyalty 
to truth in speaking and thinking of him unscientifically, any 
more than in speaking and thinking unscientifically of our 
human companions. 

This attitude may seem to allow too much scope for self- 
deception. The fear is that when we come to analyse by 
scientific methods that which we call religious experience, 
we shall find that the God whom we seem to meet in it is 
a personification of certain abstract principles. I admit that 
the application of any method which would ordinarily be 
called scientific is likely to lead to this result. But what else 
could we expect? If we confine^ourselves LtQjthe r method^ of 
physical science we shall ne"cess^Ily obtain] jfeg, ^g!MP~ 
structure of the religious experience if it has any. If we 
follow the less exact sciences they involve the same kind of 
abstraction and codifying. If our method consists in codi- 
fying, what can we possibly obtain but a code? If scientific 
method is found to reduce God to something like an ethical 
code, this is a sidelight on the nature of scientific method; 
I doubt if it throws any light on the nature of God. If the 
consideration of religious experience in the light of psycho- 
logy seems to remove from our conception of God every 
attribute that calls forth worship and devotion, it is well to 
consider whether something of the same sort has not happened 
to our human friends after psychology has analysed and 
scheduled them. 


is not necessarily to be condemned as 
of structure which is all that physical science ecognises in 

me? ~~ """ 


Let us now consider our answer to the question whether the 
nature of reality is material or spiritual or a combination of 
both. I have often indicated my dislike of the word "reality " 
which so often darkens counsel; but I state the question as it 
is commonly worded, and answer what I think is in the mind 
of the querist. 

I will first ask another question. Is the ocean composed 
of water or of waves or of both? Some of my fellow 
passengers on the Atlantic were emphatically of the opinion 
that it is composed of waves; but I think the ordinary un- 
prejudiced answer would be that it is composed of water. 
At least if we declare our belief that the nature of the ocean 
is aqueous, it is not likely that anyone will challenge us and 
assert that on the contrary its nature is undulatory, or that 
it is a dualism part aqueous and part undulatory. Similarly 
I assert that the nature of all reality is spiritual, not material 
nor a dualism of matter and spirit. The hypothesis that its 
nature can be to any degree material does not enter into my 
reckoning, because as we now understand matter, the putting 
together of the adjective "material" and the noun "nature" 
does not make sense. 

Interpreting the term material (or more strictly, physical); 
in the broadest sense as that with which we can become! 
acquainted through sensory experience of the external world, 
we recognise now that it corresponds to the waves not to 
the water of the ocean of reality. My answer does not deny 
the existence of the physical world, any more than the answer 
that the ocean is made of water denies the existence of ocean 
waves; only we do not get down to the intrinsic nature of 


things that way. Like the symbolic world of physics, a wave 
is a conception which is hollow enough to hold almost 
anything; we can have waves of water, of air, of aether, and 
(in quantum theory) waves of probability. So after physics 
has shown us the waves, we have still to determine the content 
of the waves by some other avenue of knowledge. If you 
will understand that the spiritual aspect of experience is to 
the physical aspect in the same kind of relation as the water 
to the wave form, I can leave you to draw up your own 
answer to the question propounded at the beginning of this 
section and so avoid any verbal misunderstanding. What is 
more important you will see how easily the two aspects of 
experience now dovetail together, not contesting each other's 
place. It is almost as though the modern conception of the 
physical world had deliberately left room for the reality of 
spirit and consciousness. 

In recognising only two alternatives, material and spiritual, 
we must naturally employ these terms in a very broad sense. 
We cannot suppose that the non-material substratum of the 
physical symbols has elsewhere the specialised development 
which we recognise in the substratum of the physical symbols 
which stand for ourselves. But without committing our- 
selves to any hypothetical generalisation, we can hardly do 
otherwise than name it spiritual in accordance with the one 
clue that we have as to its nature. 

To see the conception as a whole, consider how you 
yourself enter into the scheme of knowledge. By scientific 
investigation I can describe you as part of the physical 
universe, locate you in space and time, determine your 
chemical composition, and so on. This is indirect knowledge, 
for it has come to me (like all my sensory experience) 
through physical changes propagated along my nervous 
system. To give this knowledge its most precise form I have 
to use the symbols of mathematical physics and the equations 
connecting them. This does not exhaust my knowfcdge of 


you. I am convinced that associated with that portion of 
your brain, which the physiologist identifies more particularly 
as "you'*, there is something more. You are not only what 
these physical symbols describe, but also that "something to 
which truth matters" whose existence in the world of 
experience we had to admit from the beginning of our 
inquiry. I should not be lecturing to you if I were not 
convinced of this. As an inference, this knowledge of you 
is even more remote than my knowledge of your physical 
structure; for it is deduced partly from your physical mani- 
festations and behaviour, and partly from my immediate 
knowledge of what such manifestations and behaviour imply 
in my own case. But though the journey is longer, the 
destination is nearer home. For the knowledge is no longer 
of the symbolic kind; such a nature as I attribute to you is 
made up of qualities known to me in my own mind without 
the intervention of sensory mechanism. 

To what extent does this outlook involve the modern 
conceptions of physics? It is affected in this way. An 
unreflecting philosophy assumes that the nature of a table 
is "known to me in my own mind without the intervention 
of sensory mechanism". Anyone who has the task of ex- 
pounding the theory of relativity finds himself up against 
the widespread belief that the nature of space* is known in 
the mind without the intervention of sensory mechanism. 
It is due to the relativity theory and the quantum theory that 
these assumptions have been eradicated from physics, and 
replaced by the conception of symbolic knowledge which 
plays so important a part in the argument. 

* I do not add Time, because it seems to me that we have immediate 
knowledge of the time sequence in consciousness; and one of the tasks 
of physics has been to discover the relation between this immediate 
knowledge of time and our symbolic knowledge of time in the external 
wor!3. "obtained through our sensory mechanism. (See The Nature of the 
Physical World, pp. 51, tod.) 

ENPS 21 


It may be asked, Do you then believe that the same spiritual 
nature which underlies the atoms and electrons in the living 
brain pervades all atoms and electrons ? I would answer that 
it is inappropriate to speak of atoms and electrons in this 
connection. We have evidence that your consciousness is 
associated with a certain portion of your brain; but we do 
not go on to assume that a particular element of your con- 
sciousness is associated with a particular atom in your brain. 
The elements of consciousness are particular thoughts and 
feelings ; the elements of the brain cell are atoms and electrons ; 
but the two analyses do not run parallel to one another. 
Whilst therefore I contemplate a spiritual domain under- 
lying the physical world as a whole, I do not think of it as 
distributed so that to each element of time and space there 
is a corresponding portion of the spiritual background. My 
conclusion is that, although for the most part our inquiry 
into the problem of experience ends in a veil of symbols, 
there is an immediate knowledge in the minds of conscious 
beings which lifts the veil in places; and what we discern 
through these openings is of mental and spiritual nature. 
Elsewhere we see no more than the veil. 

We have travelled far from those comfortable days when, 
however ignorant we might feel as to the details of the 
construction of matter, everyone was convinced that he was 
quite familiar with its essential nature. What are my feelings, 
my thoughts? What am I myself? Mysteries too deep for 
the intellect to fathom. What is this table? Oh! Everyone 
understands that; it is just substance, commonsense reality, 
reassuringly comprehensible amid the phantasms of our 
thoughts. No. It is a commonplace reflection that we under- 
stand very little about our own minds, but it is here if anywhere 
that all knowledge begins. As for the external objects, 
remorselessly dissected by science, they are studied and 
measured, but they are never known. Our pursuit of them 
has led from solid matter to molecules, from molecules to 


sparsely scattered electric charges, from electric charges to 
waves of probability. Whither next? 

This does not lead to pure subjectivism. The physical 
object in the world of my perception is also in the world 
of your perception. There is an external world not part of 
the mind of either of us, but neutral ground wherein is 
located the basis of that experience which we hold in com- 
mon. But I think there can be no doubt that the scientist has 
a much more mystic conception of the external world than 
he had in the last century when every scientific " explana- 
tion" of phenomena proceeded on the assumption that 
nothing could be true unless an engineer could make a model 
of it. The cruder kind of materialism which sought to reduce 
everything in the universe, inorganic and organic, to a 
mechanism of fly-wheels or vortices or similar devices has 
disappeared altogether. Mechanical explanations of gravita- 
tion or electricity are laughed at nowadays. You could now 
safely hand over the human intellect to the tender mercies 
of the physicist without fear that he would discover in its 
workings a grinding of cog-wheels. But we must not make 
too much of these signs of grace in modern physical science. 
The tyranny of the engineer has been superseded by the 
tyranny of the mathematician. At least that is a view very 
widely taken. But alongside this there is a growing realisa- 
tion that the mathematician is less oppressive a master than 
the engineer, for he does not claim any insight deeper than 
his own symbols. 

In an earlier book* I have referred to the unconscious habit 
of the modern physicist of looking on the Creation as though 
it were the work of a mathematician. Perhaps the irony of 
these passages is not so evident now as it was at the time. 
I could not foresee that a few years later a colleague would 
seriously put forward the view that "from the intrinsic 
evidence of his creation, the Great Architect of the Universe 
* The Nature of the Physical World, pp. 104, 209. 



now begins to appear as a pure mathematician".* Jeans hac 
previously considered but rejected another explanation. " So 
it may be suggested, the mathematician only sees nature 
through the mathematical blinkers he has fashioned foi 

In rejecting what seems to me to be the right explanation, 
Jeans dwells on the failure of anthropomorphic theories and 
later the devices of the engineer to explain the universe, and 
he contrasts them with the success of the mathematical con- 
ception. There are two factors which, it seems to me, explain 
the comparative success of the mathematician. In the first 
place the mathematician is the professional wielder of sym- 
bols; he can deal with unknown quantities and even unknown 
operations. Clearly he is the man to help us to sift a little 
knowledge from a vast unknown. But the main reason why 
the mathematician has beaten his rivals is that we have allowed 
him to dictate the terms of die competition. The fate of every 
theory of the universe is decided by a numerical test. Does 
the sum come out right ? I am not sure that the mathematician 
understands this world of ours better than the poet and the 
mystic. Perhaps it is only that he is better at sums. 


The stress here laid on the limitations of physical science will, 
I hope, not be misunderstood by the reader. There is no 
suggestion that science has become a declining force; rather 
we obtain a clearer appreciation of the contribution which 
it is able to make, both now and in the future, to human 
development and culture. Within its own limitations physical 
science has become greatly strengthened by the changes. It 
has become more sure of its aims and perhaps less sure of 
its achievements. Since the last most bewildering revolution 
of physical theory (wave mechanics) there has been an 

* Sir James Jeans, The Mysterious Universe, p. 134. 


interval of some years during which it has been possible to 
settle down to steady progress. Recently the most striking 
developments have been on the experimental side. In quick 
succession the artificial transmutation of the elements, the 
discovery of the neutron and the discovery of the positive 
electron have startled the scientific world and opened up new 
realms for exploration. But I count this as normal prosperity 
rather than revolution. 

In contemplating the gradually developing scheme of 
scientific knowledge which never seems to reach finality in 
any direction, there are times when we are tempted to doubt 
the substantiality of our gains. Questions, which seem to 
have been settled, become unsettlec 

Nature and Nature's laws lay hid in night: 
God said, "Let Newton be!" and all was light. 
But not for long. The devil howling, "Ho! 
Let Einstein be!" restored the status quo. 

In my own subject of astronomy it is particularly difficult to 
know how far we may feel certain of our ground. So many 
conclusions have to be guarded by an "if". And it is some- 
times those results which have been most widely accepted 
that prove to have been most insecure. Finding ourselves 
unable to decide some of those simple fundamental questions, 
which to a large extent control the course of astronomical 
theory, we begin to doubt whether there has been any real 
progress. And then we realise with a start that ten years ago 
we did not know enough even to formulate the doubts that 
now beset us. I sometimes think that the progress of know- 
ledge is to be measured not by the questions that it has 
answered but by the questions that it provokes us to ask. 

In writing of the new pathways in science it is natural that 
the changes should be emphasised rather than the continuity 
with the past. It may seem that this is an age when we have 
scant respect for tradition, and are pulling to pieces all that 


our forerunners so laboriously erected. We have to show 
unsparingly the way in which the scientists of an earlier 
generation were misled by false assumptions, and the direction 
in which their conceptions of the universe have proved 
inadequate; but we utilise the positive contributions that 
they made, bringing us step by step nearer to the ideal. 
Progress has a ruthless side, but it is not an indiscriminate 
ruthlessness. We are not the less tenderly cherishing the seed 
planted by our predecessors because from time to time we 
transplant it into new soil where it may grow more freely. 
That is what a revolution in science means. When Einstein 
overthrew Newton's theory, he took Newton's plant, which 
had outgrown its pot, and transplanted it to a more open 

All this new growth of science has its roots in the past. 
If we see farther than our predecessors it is because we stand 
on their shoulders and it is not surprising if they receive a 
few kicks as we scramble up. A new generation is climbing 
on to the shoulders of the generation to which I belong; and 
so it will go on. Each phase of the scientific advance has 
contributed something that is preserved in the succeeding 
phase. That, indeed, is our ground for hope that the coming 
generation will find something worth preserving something 
that is not wholly illusory in the scientific thought of the 
Universe as it stands to-day. 

When we see these new developments in perspective they 
appear as the natural unfolding of a flower: 

For out of olde feldes, as men seith, 
Cometh al this newe corn fro yere to yere; 
And out of olde bokes, in good feith, 
Cometh al this newe science that men lere. 


A priori probability distribution, 

48, 129, 131, 249 
Absorption of radiation, 37; in 

stars, 141; in cosmic cloud, 

189, 199 

Action, atoms of, 234 
Adams, W. S., 155 
Aether, 38; mass of, 47 
Age, of sun, 165, 167; of universe, 

170, 210 
Air, 187, 205 
Alpha particles, 31, 179 
Angular momentum, uncertainty 

of, 105 
Annihilation, of matter, 165, 180; 

of positrons, 28, 180 
Anti-chance, 60, 69 
Anticommuting operators, 268, 


Anti-evolution, 54 
Archimedes, 221 
Architect of the universe, 323 
Aristarchus, 221 
Aston, F. W., 182 
Atkinson, R. D'E., 179 
Atom, structure of, 29, 46, 258; 

ionisation of, 32, 144 
Atomic number, 29 

Becoming, 53 

Beginning of the world, 58, 60, 

67, 220, 306 
Bertrand,J., 123 
Beta rays, 31, 305 
Betelgeuse, 153, 169 
Birge, R. T., 251 
Blackett, P. M. S., 28 
Body and mind, 69, 86, 313 

Bohr, N., 72; model atom, 34, 47; 

correspondence principle, 78 
Bond, W. N., 251 
Born, M., 72, 82 
Bowen, I. S., 204 
Brain and mind, 88 
Broad, C D., 75 
Broglie, L. de, 41 

Calcium, interstellar, 187, 193,201 
Canticles, 84 

Causality, law of, 74, 85, 296 
Cause and effect, 74, 296 
Cavendish experiment, 253 
Cavendish laboratory, 144 
Cepheid variables, 174, 208 
Chance coincidences, 61, 64, 89 
Chemistry, law of, 35 
Chlorine, isotopes of, 30 
Closure of space, 50, 217, 254 
Cockcroft, J. D., 160 
Coincidences as observational data, 


Coincidences, chance, 61, 64, 89 
Collisions, atomic, 145, 198 
Colour sense, 5, n 
Communal objects, 282 
Companion of Sirius, 155 
Comparison objects and standards, 


Compressibility of matter, 154, 159 
Condorcet, Marquis de, 123 
Conservation of energy, 108, 305 
Constants of Nature, primitive, 

230; numerical, 232 
Correspondence principle, 78 
Cosmic cloud, 185; density of, 

194; temperature of, 198 


Cosmic rays, 36, 164, 179 
Cosmical constant, 47, 214, 220, 

222, 230, 247 

Cosmical repulsion, 213 
Coulomb energy, 236, 240, 243 
Creation, 58, 306, 323 
Cryptogram, 8, n, 51 
Curvature of space-time, 47, 108, 

213, 315 

Cyclic scheme of knowledge, 294 
Cyclic universe, 59 

D line (sodium), 188, 201 
Data, initial, 7, 282 
Data, sense, n, 18, 284 
Degenerate matter, 158 
Dense matter in stars, 155 
Detailed balancing, principle of, 57 
Determinism, 72, 295; definitions 

of, 74; and human volition, 

86, 301 

Deuterium, 31 
Deutons, 31 
Dingle, H., 20 
Dirac, P. A. M., 216, 236, 271, 272, 

276, 292 

Disorganisation, 55 
Dissipation of energy, 66 
Distances, table of, 207 
Dogmatism, 304 
Double star, 187, 241 
Dualism, 18 

Dwarf and giant stars, 153 
Dwarfs, white, 156, 172 
Dynamical velocity, 242 

E. & O.E., 22 

E symbols, 236, 269, 276 

Eclipses, prediction of, 83 

Ego, 88 

Einstein, A., 13, 19, 164, 214, 217, 

. 297, 298, 302 
Einstein's law of gravitation, 133, 

Electromagnetic waves, 36 
Electrons, 21, 28 ; orbits of, 34, 82, 

107, 204, 258; wave nature, 
42, 101 ; mass, 243, 247; wave 
equation of, 227, 247 

Elements, number of, 30; trans- 
mutation of, 33, 1 60, 1 66, 176 
Emission of radiation, 37; in 

nebulae, 203 
Empty space, 48 
Encounters, atomic, 145, 198 
Energy, disorganisation of, 55; 
source of stellar, 143, 164; 
conservation of, 108, 305 
Energy and mass, equivalence of, 

134, 164 

Engineer and mathematician, 323 
Entropy, 55 
Evolution, 54, 58; of elements, 33, 

1 68; of stars, 169 
Examination question, 121 
Excitation of atoms, 36, 106 
Exclusion method, 120, 129, 238 
Exclusion principle (Pauli's), 23, 

35, 107, 157 
Existence, 25, 291 
Expansion of space, 215, 218 
Expansion of the universe, 67, 211 
Experience, problem of, 91, 310 
External world, 9, 45, 281, 323 

Familiar and scientific stories, 2 

Faraday, M., 40 

Fermi-Dirac statistics, 239 

Field, 39 

Field-matter theory, 41, 49 

Fine-structure constant, 232, 234, 

Finite but unbounded space, 50, 

108, 217 

Fluctuations (of entropy), 63 
Fluorescence, 202 

Fog (wave mechanics), 42, 246 

r< _ 1 ; 1 1 



Force, origin of, 213, 240 
Forms of existence, 17 
Fourth dimension, 275 
Fowler, R. H., 156 
Fractionating operators, 264 
Freedom, degrees of, 242 
Free-will, 86, 90, 301, 303 
Frequency and probability, 1 14 
Friedman, A., 213, 227 
Fiirth, R., 253 

Galaxies, 206; recession of, 209, 


Galaxy, rotation of the, 190 
Geometrisation of physics, 12 
Giant stars, 152, 172 
Globe of water, maximum, 194 
Gold standard, 81 
Gravitation, law of, 133, 194 
Group, 262 
Group-structure, 256, 274, 318 

H line (calcium), 187, 193, 201 

Hartmann, J., 187 

Heat, nature of, 55, 139; two forms 

of, 139; maintenance of sun's, 

143, 162, 164 
Heat-death, 59 
Heath, Sir T. L., 221 
Heavy hydrogen, 30 
Heavy water, 31, 264 
Heisenberg, W., 41, 46 
Heisenberg's uncertainty principle, 

45, 70, 97, 102, 248 
Helium nucleus, 3 1 ; formation of, 

167, 178; isotope of, 178 
Henry I, 230 
Hertzsprung, 152 
Hubble, E. P., 208, 213 
Humason, M. L., 212 
Hydrogen atom, 30; abundance in 

stars, 147, 149; transmutation 

of, 167, 177 
Hypersphere, 218, 252 

Hypotheses, 20, 266 

Idempotency, 263 

Ignorable coordinate, 243 

Ignorance and probability, 122, 133 

Imperfect gas, 154 

Impossible and improbable, 64, 79 

Indeterminacy of the present, 97, 

100, see Uncertainty principle 
Indeterminism of the future, 76; 

amount of, 82, 88, 101 
Indeterministic law, 80, 299 
Indifference, principle of, 122, 134 
Indistinguishable particles, 240, 251 
Inert gases, 3 5 
Inference, 9, 92; remote, 5, 280; 

retrospective, 93; system of, 


Infinity, 217 
Initial data, 7, 282 
Instability of atoms, 33, 178; of 

stars, 174; of universe, 220 
Integers, displacement of, 23 
Interaction, 238 

Interchange, energy of, 240, 243 
Interstellar matter, see Cosmic 


Interval, 275 

Inverse probability, 125, 127 
lonisation, 32, 144; of interstellar 

matter, 201 

Irreducible energy, 108 
Irreversible processes, 66 
Isotopes, 30; of potassium, 95, 104 
Isotropy, origin of, 133 

Jabberwocky, 256 
Jeans, SirJ. H., 136,324 
Joad, C. E. M., 281, 288 
Jumps, orbit, 36, 106, 204, 258 

K group of electrons, 34, 37 
K line (calcium), 187, 193, 201 
Kelvin, Lord, 66, 162 

330 INDEX 

Knight's moves, 259 
Knowable universe, 104 
Rummer's quartic surface, 271 

L group of electrons, 34, 37 
Lane,J. H., 135, 138, 152 
Laplace, P. S., 74, no, 195 
Larmor, Sir J., 41, 182 
Laws of Nature, 8 ; primary and 

secondary, 80; causal and 

statistical, 80, 299 
Lemaitre, G., 213, 220, 227 
Life, 310, 312 
Light, see Radiation 
Limitations of physical science, 


Lindemann, F. A., 106 
Lorentz, H. A., 41 
Luminosity of stars, 136, 141, 148, 


Macroscopic and microscopic 
theories, 20, 131, 244 

Main series, 172 

Man, 310, 316 

Mass, equivalent to energy, 134, 
164; origin of, 108, 249; of 
electron and proton, 230, 243, 
247; of sun, 138; of universe, 
221, 248 

Mass-defect, 33, 167 

Mass-luminosity relation, 153 

Mass-ratio, 232, 243, 247, 250 

Mathematician and engineer, 323 

Maxwell,]. .,68,233 

Mental realm, 283 

Meteors, 164, 202 

Metrical field, 39; tensor, 131, 252 

Microscopic and macroscopic 
theories, 20, 131, 244 

Mind and time-direction, 52; 
and entropy, 69; and deter- 
minism, 86; and sense-data, 

Minkowski, H., 275 

Models, 266 

Momentum, identified with curva- 
ture, 47, 108 ; uncertainty of, 
101 ; angular, 105 ; of indis- 
tinguishable particles, 242 

Multillions, 60 

Nebulae, gaseous, 184, 196, 202; 
dark obscuring, 184, 202 

Nebulae, spiral, 206; recession of, 
209, 220 

Nebulium, 203 

Negatron, 28, 181 

Nerve messages, 3, 7 

Neutral realm, 283 

Neutron, 31, 32, 181; in stars, 151 

New statistics, 157, 239 

Nitrogen, 187, 205 

Non-technical writing, aim of, 279 

Not-hydrogen, 30, 147, 167 

Nucleus, atomic, 29; structure of, 
32; bombardment of, 33, 176; 
mass-defect, 33 

Number of particles in the uni- 
verse, 221, 248, 250, 252 

Objective reality, i, 45, 46, 104, 


Obscuring nebulosity, 184, 202 
Observation and theory, 211 
Observer, ideal, 13 
Occhialini, G. P. S., 28 
Oort, J. H., 191 
Opacity of stellar matter, 141, 

Operators (mathematical), 260, 

263, 267 
Orbits of electrons, 34, 82, 107, 

204, 258 

Organisation, 55 
Organism, 312 
Oxygen, 187, 205 
Ozone layer, 193 



Paired quantities, 100 
Particles, high-speed, 33, 176; in- 
distinguishable, 240 
Pauli, W., 23, 35, 157 
Pearce, J. A., 190 
Periodic table of elements, 35 
Permutation variable, 243 
Perseus, cluster in, 190 
Personification, 319 
Philosophers, 5, 24, 281 
Photo-electric cell, 14 
Photo-electric effect, 37, 199 
Photons, 38, 145 
Physical science, limitations of, 

315, 324 

Planck, M., 72, 295, 299 
Planck's constant (A), 37, 101,231, 


Plaskett,]. S., 188, 190 
Pluto, 29 

Pointer-readings, 13, 15, 256 
Popular writing, aim of, 279; 

dogmatism in, 304 
Positrons, 28, 32, 180 
Potassium, isotopes of, 95, 104 
Predetermination, 76, 84, 93 
Pressure in sun, 139; effect on 

matter, 159 
Primary aim of science, 77; 

system of law, 80 
Principle, correspondence, 78 
Principle of detailed balancing, 57 
Principle, exclusion, 23, 35, 107, 


Principle of indifference, 122, 134 
Principle of least action, 234 
Principle of relativity, 102, 223 
Principle of uncertainty (or in- 
determinacy), 45, 70, 97, 102, 
Probability, 42, 64, no; universe 

of, 128 

Probability distributions, 48, 129, 
131, 238, 245, 249, 252 

Protons, 28 ; in sun, 145 ; transmu- 
tation of, 177; mass of, 247 
Psi (0), 44, no 
Pulsating stars, 136, 174 
Purity, 264 
Purpose, 60, 69 
Pythagoras's theorem, 273, 275 

Quantum, 37, 101, 235 
Quantum laws, 37, 199 
Quantum theory, and relativity 
theory, 20, 131, 244; and 
wave mechanics, 41 
Quantum theory, criticisms of, 

224, 240, 245 
Quotations from 

Advertisement, 92 

Archimedes, 221, 222 

Bohr, N., 72 

Born, M., 72, 82 

Braithwaite, R. B., 278 

Broad, C. D., 75 

Chaucer, 326 

Dickens, 278 

Einstein, A., 297, 298 

Holmes, O. W., 62 

Jeans, Sir J. H., 323 

Joad, C. E. M., 289, 294 

Keats, 184 

Laplace, P. S., 74, no, 194 

Lewis Carroll, 219, 255 

Milton, 206 

Omar Khayyam, 75 

Planck, M., 72, 298, 299, 300,301 

Poincare", H., i 

Rossetti, D. G., 315 

Russell, Earl, 255, 306, 307 

Rutherford, Lord, 297 

Shakespeare, 27, 135, 19? 

Stace, W. T., 283 

Swift, 160 

Unknown authors, 50, 62, 325 

Virgil, 229 

Weyl, H., 72, 309 

332 INDEX 

Quotations from (cont.) 
Whitehead, A. N., 48 
Wordsworth, 205 

Radiant heat, 139 

Radiation, 36; inside a star, 141, 

145 ; amount of solar, 165 ; in 

interstellar space, 198; of 

gaseous nebulae, 202 
Radio-activity, 33, 83, 95, i4, 166 
Realist philosophy, 281, 288, 289 
Reality, 25, 291, 319 
Reasoning, respect for, 91, 317 
Recession of spiral nebulae, 209, 220 
Recurrencies, 8, 284 
Redundancy of sense data, n 
Relativity theory, 18, 102, 130,156, 

Relativity theory and quantum 

theory, 20, 131, 244 
Religion, 306, 317 
Repulsion, cosmical, 213, 220 
Responsibility, 89, 311 
Retrospective characters, 94, 98 
Retrospective inference, 93 
Revolutions of science, 326 
Right-thinking person, 112, 119 
Rigour, 257, 280 
Rotation, of stars, 148; of sun, 187; 

of the galaxy, 190 
Russell, B. (Earl Russell), 101, 255, 

257, 305 

Russell, H. N., 152 
Rutherford, Lord, 160, 297 

Samuel, Sir H., 295 
Satellite electrons, 29 
Schrodinger, E., 41 
Schwarzschild, K., 194 
Scientific and familiar stories, 2 
Second law of thermodynamics, 

55; in closed space, 66; for 

conscious beings, 68 
Secondary aim of science, 77 

Secondary law, 80, 128 

Selective operators, 263 

Sieve, 267 

Signpost for time, 52, 67 

Sirius, companion of, 155 

Sitter, W. de, 212 

Slipher, V. M., 188, 212 

Sodium, interstellar, 188, I93 201 

Sorting demon, 68 

Source of star's heat, 142, 150, 162, 

Space, relation to aether, 39; not 

empty, 48 ; closure of, 50, 217, 

254; interstellar, 184; expan- 
sion of, 215, 218; distinction 

from time, 276 
Space-like operators, 277 
Spectrum, 37, 137, 193; of gaseous 

nebulae, 202; red-shift of, 212 
Spherical space, 51, 218 
Spin of electron, 41, 276 
Spiral nebulae, 206; recession of, 

209, 220 
Spiritual nature of Man, 311; of 

reality, 319, 322 
Stace, W. T., 281, 283 
Standards of length, 230; relativity 

of, 215, 224 

Stars, general characteristics of, 136 
Statistical law, 80, 128, 299 
Statistics, new, 157, 239 
Stefan's constant, 233 
Stromgren, B., 147 
Structure, 16, 256, 262 
Struve, O., 189 
Subatomic energy, 143 ; utilisation 

of, 163; two forms of, 167; 

conditions of release, 175 
Subjective element in physics, 45, 

104, 292 
Sun, dimensions of, 138 ; energy of, 

164; age of, 165, 167; rotation 

of, 187 
Super-dense matter, 156 

Super-mathematicians, 257, 261, 

Temperature of stars, surface, 136; 

internal, 138; confirmation of, 


Temperature of cosmic cloud, 198 
Theory and observation, 211 
Thermodynamic equilibrium, 57, 

Thermodynamics, 76; second law 

of, 55, 66, 68 
Thingless space, 48 
Time, signpost for, 52, 67; distinc- 
tion from space, 276 
Time lag of energy liberation, 175, 


Time-scale, 67, 165, 167, 171, 210 
Transmutation of elements, 33, 

160, 166, 176 
Truth, 311 

Uncertainty principle, 45, 70, 97, 
102, 248 

Unification of physics, n, 233 

Unification of relativity and quan- 
tum theories, 47, 108, 130, 244 

Universe, expansion of, 211; as 
comparison standard, 226, 

247; mass of, 221, 248 
number of particles in, 221 
248, 250, 252 
Unknowable quantities, 98 

Vacuum, 49, 153, *97 

Velocity, uncertainty of, 101 

for indistinguishable particles 

Volition, 88 

Walton, E. T. S., 160 

Wave equation of electron, 226 


Wave functions, 224, 226, 246 
Wave mechanics, 41, 107, 128, 148 


Wave packets, 101 
Waves and substance, 319 
Waves, electromagnetic, 36, se< 


Weyl, H., 72, 215, 309 
White dwarfs, 156, 172 
Whitehead, A. N., 48 
Wilson, C. T. R., 28 

X rays, 36, 37; in stars, 140, 145 
Yard-stick and yard, 224